GIÁO TRÌNH TOÁN 9

["GI\u00c1O TR\u00ccNH TO\u00c1N 9 7) S = 5 v\u00e0 P = \u22126; 9) S = \u22121 v\u00e0 P = \u221220; 8) S = \u22123 v\u00e0 P = \u221218; 10) S = 3 v\u00e0 P = \u221210. B\u00e0i 2. Hai s\u1ed1 c\u00f3 t\u1ed5ng b\u1eb1ng 29, t\u00edch b\u1eb1ng 204. T\u00ecm hai s\u1ed1 \u0111\u00f3. B\u00e0i 3. N\u1ebfu hai s\u1ed1 u v\u00e0 v c\u00f3 u + v = 32, u.v = 231 th\u00ec u v\u00e0 v l\u00e0 bao nhi\u00eau? B\u00e0i 4. T\u00ecm hai s\u1ed1 a v\u00e0 b bi\u1ebft a + b = 29 v\u00e0 a.b = 198. | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M \u221a\u221a B\u00e0i 5. Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m l\u00e0: x1 = \u2212 3; x2 = 2 + 3. \u221a\u221a B\u00e0i 6. Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m x1 = 1 \u2212 5; x2 = 1 + 5. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 6 Ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 B\u00e0i 1. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t: 1) x2 + x + m = 0; 7) \u22123x2 + x + 4m + 1 = 0; 2) x2 \u2212 x + m \u2212 1 = 0; 8) mx2 + x + 1 = 0; 3) x2 + 3x \u2212 m + 2 = 0; 9) mx2 \u2212 2x + 3 = 0; 4) \u2212x2 + 5x \u2212 3m + 4 = 0; 10) \u2212mx2 \u2212 5x + 2 = 0; 5) 2x2 \u2212 3x + m \u2212 1 = 0; 11) (m + 1) x2 + 2x + 1 = 0; 6) x2 \u2212 2mx + m2 \u2212 m = 0; 12) (6 \u2212 3m) x2 + 2x \u2212 1 = 0. B\u00e0i 2. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 nghi\u1ec7m k\u00e9p: 7) x2 + 2 (m + 1) x + m2 \u2212 3 = 0; 1) x2 \u2212 x + m = 0; 2) x2 + 2x \u2212 2m = 0; 8) x2 \u2212 mx + m \u2212 1 = 0; 3) x2 \u2212 2x \u2212 2m + 1 = 0; 9) 2x2 + mx + m \u2212 2 = 0; 4) x2 + 6x + m \u2212 3 = 0; 10) x2 + (m + 5) x \u2212 m \u2212 6 = 0; 5) x2 \u2212 2mx + m2 \u2212 m + 1 = 0; 11) x2 \u2212 (m \u2212 2) x \u2212 2m = 0; 6) x2 + 2mx + m2 + 2m \u2212 3 = 0; 12) mx2 \u2212 2mx + m + 3 = 0. B\u00e0i 3. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh v\u00f4 nghi\u1ec7m: 4) x2 + 3x + m + 1 = 0; 1) x2 \u2212 x + m = 0; 2) 2x2 + x \u2212 m = 0; 5) x2 + 3x + m \u2212 2 = 0; 3) \u2212x2 + 2x + m \u2212 1 = 0; 6) x2 \u2212 4x \u2212 2m + 1 = 0; | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 45","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m 7) 3x2 \u2212 2x \u2212 m \u2212 3 = 0; 10) mx2 \u2212 x + 1 = 0; 8) \u22125x2 + x + m \u2212 2 = 0; 11) (2m + 1) x2 \u2212 x + 2 = 0; 9) \u2212x2 + 4x \u2212 2m + 1 = 0; 12) (1 \u2212 2m) x2 \u2212 2x + 4 = 0. B\u00e0i 4. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh sau c\u00f3 nghi\u1ec7m: 7) \u22122x2 \u2212 3x + 2 \u2212 m = 0; 1) x2 \u2212 2x + m = 0; 2) x2 \u2212 3x + 2m = 0; 8) 3x2 \u2212 x + m \u2212 1 = 0; | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT 3) x2 + 2x + m \u2212 1 = 0; 9) mx2 + x \u2212 1 = 0; 4) x2 \u2212 4x + 1 \u2212 2m = 0; 10) mx2 + 2x + 1 = 0; 5) \u2212x2 + 3x \u2212 m + 3 = 0; 11) (m \u2212 1) x2 + 3x + 1 = 0; 6) \u22122x2 + 3x + 2m \u2212 1 = 0; 12) (2 \u2212 m) x2 + 3x + 2 = 0. B\u00e0i 5. Ch\u1ee9ng minh ph\u01b0\u01a1ng tr\u00ecnh lu\u00f4n c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t: 1) x2 \u2212 mx \u2212 1 = 0; 8) x2 \u2212 (m + 1) x + m \u2212 1 = 0; 2) x2 \u2212 2mx \u2212 3 = 0; 9) \u22122x2 + (m \u2212 2) x + m \u2212 3 = 0; 3) 2x2 + mx + m \u2212 3 = 0; 10) \u22124x2 + (2m \u2212 1) x + m + 1 = 0; 4) x2 \u2212 4mx + 4m2 \u2212 1 = 0; 11) x2 \u2212 (m + 1) x \u2212 2m \u2212 7 = 0; 5) 2x2 + 3mx + 3 (m \u2212 1) = 0; 12) x2 + (2m + 5) x + m2 \u2212 5m = 0; 6) \u2212x2 + (m + 1) x + 2 = 0; 13) x2 + (2m + 5) x + m2 + 3m \u2212 4 = 0; 7) x2 \u2212 (2m + 1) x \u2212 2 = 0; 14) x2 \u2212 2 (m + 1) x + m2 + 2m \u2212 1 = 0. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 7 Gi\u1ea3i b\u00e0i to\u00e1n b\u1eb1ng c\u00e1ch l\u1eadp ph\u01b0\u01a1ng tr\u00ecnh I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: C\u00e1c b\u01b0\u1edbc gi\u1ea3i b\u00e0i to\u00e1n b\u1eb1ng c\u00e1ch l\u1eadp ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc hai: B\u01b0\u1edbc 1: Ch\u1ecdn \u1ea9n, \u0111\u1eb7t \u0111\u01a1n v\u1ecb v\u00e0 \u0111i\u1ec1u ki\u1ec7n cho \u1ea9n. B\u01b0\u1edbc 2: Bi\u1ec3u di\u1ec5n c\u00e1c \u0111\u1ea1i l\u01b0\u1ee3ng ch\u01b0a bi\u1ebft theo \u1ea9n. B\u01b0\u1edbc 3: L\u1eadp ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc hai m\u1ed9t \u1ea9n. B\u01b0\u1edbc 4: Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh, so s\u00e1nh v\u1edbi \u0111i\u1ec1u ki\u1ec7n r\u1ed3i k\u1ebft lu\u1eadn. 46 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M II. B\u00e0i t\u1eadp: B\u00e0i 1. Cho tam gi\u00e1c vu\u00f4ng c\u00f3 \u0111\u1ed9 d\u00e0i c\u1ea1nh huy\u1ec1n l\u00e0 13cm, hai c\u1ea1nh g\u00f3c vu\u00f4ng h\u01a1n k\u00e9m nhau 7cm. T\u00ednh \u0111\u1ed9 d\u00e0i m\u1ed7i c\u1ea1nh g\u00f3c vu\u00f4ng. B\u00e0i 2. M\u1ed9t khu v\u01b0\u1eddn h\u00ecnh ch\u1eef nh\u1eadt c\u00f3 chi\u1ec1u d\u00e0i l\u1edbn h\u01a1n chi\u1ec1u r\u1ed9ng 5m, di\u1ec7n t\u00edch b\u1eb1ng 300m2. T\u00ednh chi\u1ec1u d\u00e0i v\u00e0 chi\u1ec1u r\u1ed9ng c\u1ee7a khu v\u01b0\u1eddn. B\u00e0i 3. T\u00ednh c\u00e1c c\u1ea1nh g\u00f3c vu\u00f4ng c\u1ee7a m\u1ed9t tam gi\u00e1c vu\u00f4ng bi\u1ebft chu vi tam gi\u00e1c b\u1eb1ng 40cm, c\u1ea1nh huy\u1ec1n d\u00e0i 17cm. B\u00e0i 4. T\u00ecm chi\u1ec1u r\u1ed9ng v\u00e0 chi\u1ec1u d\u00e0i c\u1ee7a h\u00ecnh ch\u1eef nh\u1eadt. Bi\u1ebft chi\u1ec1u d\u00e0i h\u01a1n chi\u1ec1u r\u1ed9ng 10cm v\u00e0 di\u1ec7n t\u00edch l\u00e0 600cm2. B\u00e0i 5. M\u1ed9t m\u1ea3nh \u0111\u1ea5t h\u00ecnh ch\u1eef nh\u1eadt c\u00f3 chi\u1ec1u d\u00e0i h\u01a1n chi\u1ec1u r\u1ed9ng 7cm v\u00e0 di\u1ec7n t\u00edch l\u00e0 800cm2. T\u00ecm chi\u1ec1u d\u00e0i v\u00e0 chi\u1ec1u r\u1ed9ng c\u1ee7a m\u1ea3nh \u0111\u1ea5t. B\u00e0i 6. T\u00ecm hai s\u1ed1 bi\u1ebft r\u1eb1ng s\u1ed1 th\u1ee9 nh\u1ea5t nhi\u1ec1u h\u01a1n 5 l\u1ea7n s\u1ed1 th\u1ee9 hai l\u00e0 5 \u0111\u01a1n v\u1ecb v\u00e0 hi\u1ec7u b\u00ecnh ph\u01b0\u01a1ng c\u1ee7a ch\u00fang l\u00e0 351. B\u00e0i 7. M\u1ed9t khu v\u01b0\u1eddn h\u00ecnh ch\u1eef nh\u1eadt c\u00f3 chi\u1ec1u d\u00e0i g\u1ea5p 3 l\u1ea7n chi\u1ec1u r\u1ed9ng v\u00e0 c\u00f3 di\u1ec7n t\u00edch l\u00e0 507cm2. T\u00ecm chu vi khu v\u01b0\u1eddn. B\u00e0i 8. M\u1ed9t h\u00ecnh ch\u1eef nh\u1eadt c\u00f3 t\u1ed5ng chi\u1ec1u d\u00e0i v\u00e0 chi\u1ec1u r\u1ed9ng l\u00e0 34cm, di\u1ec7n t\u00edch l\u00e0 60cm2. T\u00ecm chi\u1ec1u d\u00e0i v\u00e0 chi\u1ec1u r\u1ed9ng c\u1ee7a h\u00ecnh ch\u1eef nh\u1eadt. B\u00e0i 9. M\u1ed9t m\u1ea3nh \u0111\u1ea5t h\u00ecnh ch\u1eef nh\u1eadt c\u00f3 chi\u1ec1u d\u00e0i g\u1ea5p 3 l\u1ea7n chi\u1ec1u r\u1ed9ng. N\u1ebfu gi\u1ea3m chi\u1ec1u d\u00e0i v\u00e0 chi\u1ec1u r\u1ed9ng \u0111i 4m th\u00ec di\u1ec7n t\u00edch gi\u1ea3m 208m2. T\u00ecm chi\u1ec1u d\u00e0i v\u00e0 chi\u1ec1u r\u1ed9ng m\u1ea3nh \u0111\u1ea5t. B\u00e0i 10. M\u1ed9t m\u1ea3nh \u0111\u1ea5t h\u00ecnh ch\u1eef nh\u1eadt c\u00f3 chi\u1ec1u d\u00e0i h\u01a1n chi\u1ec1u r\u1ed9ng 4m. N\u1ebfu t\u0103ng chi\u1ec1u r\u1ed9ng 2 l\u1ea7n v\u00e0 gi\u1ea3m chi\u1ec1u d\u00e0i 6m th\u00ec di\u1ec7n t\u00edch t\u0103ng th\u00eam 48m2. T\u00ecm c\u00e1c k\u00edch th\u01b0\u1edbc c\u1ee7a m\u1ea3nh \u0111\u1ea5t. B\u00e0i 11. M\u1ed9t khu \u0111\u1ea5t h\u00ecnh ch\u1eef nh\u1eadt c\u00f3 chu vi 210m. Xung quanh khu \u0111\u1ea5t, ng\u01b0\u1eddi ta l\u00e0m m\u1ed9t l\u1ed1i \u0111i r\u1ed9ng 2m, v\u00ec v\u1eady di\u1ec7n t\u00edch \u0111\u1ea5t c\u00f2n l\u1ea1i \u0111\u1ec3 tr\u1ed3ng tr\u1ecdt l\u00e0 2296m2. T\u00ednh k\u00edch th\u01b0\u1edbc c\u1ee7a khu \u0111\u1ea5t. B\u00e0i 12. Qu\u00e3ng \u0111\u01b0\u1eddng AB d\u00e0i 270km. Hai \u00f4 t\u00f4 kh\u1edfi h\u00e0nh c\u00f9ng m\u1ed9t l\u00fac \u0111i t\u1eeb A \u0111\u1ebfn B. \u00d4 t\u00f4 th\u1ee9 nh\u1ea5t ch\u1ea1y nhanh h\u01a1n \u00f4 t\u00f4 th\u1ee9 hai 12km\/h, n\u00ean \u0111\u1ebfn tr\u01b0\u1edbc \u00f4 t\u00f4 th\u1ee9 hai 40 ph\u00fat. T\u00ednh v\u1eadn t\u1ed1c c\u1ee7a m\u1ed7i \u00f4 t\u00f4. B\u00e0i 13. Hai ng\u01b0\u1eddi \u0111i xe \u0111\u1ea1p c\u00f9ng xu\u1ea5t ph\u00e1t m\u1ed9t l\u00fac \u0111i t\u1eeb A \u0111\u1ebfn B d\u00e0i 30km, v\u1eadn t\u1ed1c c\u1ee7a h\u1ecd h\u01a1n k\u00e9m nhau 3km\/h n\u00ean \u0111\u1ebfn B s\u1edbm mu\u1ed9n h\u01a1n nhau 30 ph\u00fat. T\u00ednh v\u1eadn t\u1ed1c c\u1ee7a m\u1ed7i ng\u01b0\u1eddi. B\u00e0i 14. M\u1ed9t ng\u01b0\u1eddi \u0111i t\u1eeb t\u1ec9nh A \u0111\u1ebfn t\u1ec9nh B c\u00e1ch nhau 78km, sau \u0111\u00f3 1 gi\u1edd ng\u01b0\u1eddi th\u1ee9 hai \u0111i t\u1eeb t\u1ec9nh B \u0111\u1ebfn t\u1ec9nh A hai ng\u01b0\u1eddi g\u1eb7p nhau t\u1ea1i \u0111\u1ecba \u0111i\u1ec3m C c\u00e1ch B 36km. T\u00ednh th\u1eddi gian m\u1ed7i ng\u01b0\u1eddi \u0111\u00e3 \u0111i t\u1eeb l\u00fac kh\u1edfi h\u00e0nh \u0111\u1ebfn l\u00fac g\u1eb7p nhau, bi\u1ebft v\u1eadn t\u1ed1c ng\u01b0\u1eddi th\u1ee9 hai l\u1edbn h\u01a1n ng\u01b0\u1eddi th\u1ee9 nh\u1ea5t l\u00e0 | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 47","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m 4km\/h. | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT B\u00e0i 15. M\u1ed7i xe ph\u1ea3i ch\u1edf 168 t\u1ea5n th\u00f3c. N\u1ebfu t\u0103ng th\u00eam 6 xe v\u00e0 ch\u1edf th\u00eam 12 t\u1ea5n th\u00f3c th\u00ec m\u1ed7i xe ch\u1edf nh\u1eb9 h\u01a1n l\u00fac \u0111\u1ea7u l\u00e0 1 t\u1ea5n. H\u1ecfi l\u00fac \u0111\u1ea7u m\u1ed7i \u0111\u1ed9i c\u00f3 bao nhi\u00eau xe. B\u00e0i 16. M\u1ed9t ca n\u00f4 xu\u00f4i d\u00f2ng 42km r\u1ed3i ng\u01b0\u1ee3c tr\u1edf l\u1ea1i 20km h\u1ebft t\u1ed5ng c\u1ed9ng 5 gi\u1edd. Bi\u1ebft v\u1eadn t\u1ed1c c\u1ee7a d\u00f2ng ch\u1ea3y l\u00e0 2km\/h. T\u00ednh v\u1eadn t\u1ed1c c\u1ee7a ca n\u00f4 l\u00fac d\u00f2ng n\u01b0\u1edbc y\u00ean l\u1eb7ng. B\u00e0i 17. H\u00e0 N\u1ed9i c\u00e1ch Nam \u0110\u1ecbnh 90km. Hai \u00f4 t\u00f4 kh\u1edfi h\u00e0nh \u0111\u1ed3ng th\u1eddi, m\u1ed9t xe \u0111i t\u1eeb H\u00e0 N\u1ed9i, xe kia \u0111i t\u1eeb Nam \u0110\u1ecbnh v\u00e0 \u0111i ng\u01b0\u1ee3c chi\u1ec1u nhau. Sau 1 gi\u1edd, ch\u00fang g\u1eb7p nhau, ti\u1ebfp t\u1ee5c \u0111i, xe th\u1ee9 hai t\u1edbi H\u00e0 N\u1ed9i tr\u01b0\u1edbc xe th\u1ee9 nh\u1ea5t t\u1edbi Nam \u0110\u1ecbnh l\u00e0 27 ph\u00fat. T\u00ednh v\u1eadn t\u1ed1c m\u1ed7i xe. B\u00e0i 18. M\u1ed9t \u00f4 t\u00f4 d\u1ef1 \u0111\u1ecbnh \u0111i t\u1eeb A \u0111\u1ebfn B m\u1ea5t 5 gi\u1edd. Nh\u01b0ng khi \u0111i \u0111\u01b0\u1ee3c 56km n\u00f3 d\u1eebng l\u1ea1i 10 ph\u00fat. \u0110\u1ec3 \u0111\u1ebfn B \u0111\u00fang th\u1eddi gian d\u1ef1 t\u00ednh, \u00f4 t\u00f4 ph\u1ea3i t\u0103ng v\u1eadn t\u1ed1c th\u00eam 2km\/h. T\u00ednh kho\u1ea3ng c\u00e1ch AB? B\u00e0i 19. Hai \u0111\u1ed9i c\u00f4ng nh\u00e2n c\u00f9ng l\u00e0m m\u1ed9t qu\u00e3ng \u0111\u01b0\u1eddng th\u00ec 12 ng\u00e0y xong vi\u1ec7c. N\u1ebfu m\u1ed9t \u0111\u1ed9i l\u00e0m m\u1ed9t m\u00ecnh h\u1ebft n\u1eeda c\u00f4ng vi\u1ec7c, r\u1ed3i \u0111\u1ed9i th\u1ee9 hai ti\u1ebfp t\u1ee5c m\u1ed9t m\u00ecnh l\u00e0m n\u1ed1t ph\u1ea7n vi\u1ec7c c\u00f2n l\u1ea1i th\u00ec h\u1ebft t\u1ea5t c\u1ea3 25 ng\u00e0y. H\u1ecfi m\u1ed7i \u0111\u1ed9i l\u00e0m m\u1ed9t m\u00ecnh th\u00ec bao l\u00e2u xong vi\u1ec7c? B\u00e0i 20. Ch\u00e0o m\u1eebng ng\u00e0y Gi\u1ea3i Ph\u00f3ng Mi\u1ec1n Nam (30 th\u00e1ng 4), hai ph\u00e2n x\u01b0\u1edfng c\u01a1 kh\u00ed thi \u0111ua s\u1ea3n xu\u1ea5t. M\u1ed7i ph\u00e2n x\u01b0\u1edfng ph\u1ea3i l\u00e0m 240 s\u1ea3n ph\u1ea9m trong m\u1ed9t th\u1eddi gian quy \u0111\u1ecbnh. M\u1ed7i ng\u00e0y ph\u00e2n x\u01b0\u1edfng I s\u1ea3n xu\u1ea5t nhi\u1ec1u h\u01a1n ph\u00e2n x\u01b0\u1edfng II l\u00e0 8 s\u1ea3n ph\u1ea9m v\u00e0 \u0111\u00e3 ho\u00e0n th\u00e0nh c\u00f4ng vi\u1ec7c s\u1edbm h\u01a1n th\u1eddi gian quy \u0111\u1ecbnh l\u00e0 3 ng\u00e0y v\u00e0 s\u1edbm h\u01a1n ph\u00e2n x\u01b0\u1edfng II l\u00e0 1 ng\u00e0y. H\u1ecfi th\u1eddi gian quy \u0111\u1ecbnh l\u00e0 bao nhi\u00eau ng\u00e0y? \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 8 T\u01b0\u01a1ng giao \u0111\u1ed3 th\u1ecb B\u00e0i 1. V\u1ebd \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = x2 v\u00e0 y = \u22124x + 5 tr\u00ean c\u00f9ng m\u1ed9t h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9. B\u00e0i 2. T\u00ecm t\u1ecda \u0111\u1ed9 giao \u0111i\u1ec3m c\u1ee7a (P ) : y = x2 v\u00e0 y = \u22124x + 5 b\u1eb1ng ph\u00e9p t\u00ednh. B\u00e0i 3. Cho Parabol (P ) : y = 2x2 v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng (d) : y = 2x + 3. T\u00ecm t\u1ecda \u0111\u1ed9 giao \u0111i\u1ec3m c\u1ee7a hai h\u00e0m s\u1ed1 tr\u00ean b\u1eb1ng ph\u00e9p t\u00ednh. B\u00e0i 4. T\u00ecm giao \u0111i\u1ec3m c\u1ee7a parabol (P ) : y = 2x2 v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng (d) : y = 5x \u2212 3 b\u1eb1ng ph\u00e9p t\u00ednh. B\u00e0i 5. Cho Parabol (P ) : y = x2 v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng (d) : y = 4x \u2212 3. T\u00ecm t\u1ecda \u0111\u1ed9 giao \u0111i\u1ec3m c\u1ee7a hai 48 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 h\u00e0m s\u1ed1 tr\u00ean b\u1eb1ng ph\u00e9p t\u00ednh. B\u00e0i 6. Cho Parabol (P ) : y = x2 v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng (d) : y = \u22123x + 5. T\u00ecm t\u1ecda \u0111\u1ed9 giao \u0111i\u1ec3m c\u1ee7a hai h\u00e0m s\u1ed1 tr\u00ean b\u1eb1ng ph\u00e9p t\u00ednh. B\u00e0i 7. T\u00ecm giao \u0111i\u1ec3m c\u1ee7a parabol (P ) : y = 2x2 v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng (d) : y = 5x \u2212 3 b\u1eb1ng ph\u00e9p t\u00ednh. | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M B\u00e0i 8. Cho (P ) : y = \u2212 1 x2 v\u00e0 (d) : y = x + m. T\u00ecm m \u0111\u1ec3 (d) c\u1eaft (P ) t\u1ea1i hai \u0111i\u1ec3m ph\u00e2n bi\u1ec7t. 2 B\u00e0i 9. Cho (P ) : y = 1 x2 v\u00e0 (d) : y = x \u2212 m. T\u00ecm m \u0111\u1ec3 (d) v\u00e0 (P ) ti\u1ebfp x\u00fac nhau, t\u00ecm t\u1ecda \u0111\u1ed9 4 ti\u1ebfp \u0111i\u1ec3m. B\u00e0i 10. Cho (P ) : y = \u2212 1 x2 v\u00e0 (d) : y = mx + 1. T\u00ecm m \u0111\u1ec3 (d) c\u1eaft (P ) t\u1ea1i hai \u0111i\u1ec3m ph\u00e2n bi\u1ec7t. 4 B\u00e0i 11. Cho (P ) : y = 2x2 v\u00e0 (d) : y = mx \u2212 m + 2. T\u00ecm m \u0111\u1ec3 (d) v\u00e0 (P ) c\u00f3 hai \u0111i\u1ec3m chung ph\u00e2n bi\u1ec7t. B\u00e0i 12. Cho (P ) : y = 1 x2 v\u00e0 (d) : y = mx \u2212 2m \u2212 1. T\u00ecm m \u0111\u1ec3 (d) v\u00e0 (P ) ti\u1ebfp x\u00fac nhau, t\u00ecm 4 t\u1ecda \u0111\u1ed9 ti\u1ebfp \u0111i\u1ec3m. B\u00e0i 13. Cho (P ) : y = \u2212 3 x2 v\u00e0 (d) : y = 3 T\u00ecm m \u0111\u1ec3 (d) c\u1eaft (P ) t\u1ea1i hai \u0111i\u1ec3m ph\u00e2n bi\u1ec7t. x + 2m. 42 B\u00e0i 14. Cho (P ) : y = 1 x2 v\u00e0 (d) : y = m (x + 1) , m \u0338= 0. T\u00ecm m \u0111\u1ec3 (d) ti\u1ebfp x\u00fac (P ), t\u00ecm t\u1ecda 2 \u0111\u1ed9 ti\u1ebfp \u0111i\u1ec3m. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 9 \u00d4n t\u1eadp ch\u01b0\u01a1ng 4 B\u00e0i 1. Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh: 3) x4 \u2212 5x2 \u2212 6 = 0; 1) x2 \u2212 3x + 4 = 0; 2) x2 \u2212 4x + 4 = 0; 4) x4 + 6x2 + 9 = 0. B\u00e0i 2. Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m c\u1ee7a (P ) : y = x2 v\u00e0 (d) : y = 2x \u2212 1. B\u00e0i 3. Kh\u00f4ng gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh h\u00e3y t\u00ednh t\u1ed5ng v\u00e0 t\u00edch c\u00e1c nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh x2\u22128x\u22127 = 0. B\u00e0i 4. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh x2 + 3x \u2212 m = 0 c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t. Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh: 1) x2 \u2212 3x + 4 = 0; 3) 2x2 + 1 2 \u2212 3 2x2 + 1 \u2212 4 = 0; 2) x4 \u2212 5x2 \u2212 6 = 0; 4) \u2212 3x2 + 5 2 + 15x2 + 19 = 0. B\u00e0i 5. Cho ph\u01b0\u01a1ng tr\u00ecnh x2 + 5x \u2212 2m + 4 = 0. T\u00ecm m \u0111\u1ec3: 1) Ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m tr\u00e1i d\u1ea5u; 2) Ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m th\u1ecfa x21 + x22 = 5. | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 49","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m B\u00e0i 6. Cho ph\u01b0\u01a1ng tr\u00ecnh x4 \u2212 2x2 \u2212 2m + 1 = 0. 1) T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 4 nghi\u1ec7m ph\u00e2n bi\u1ec7t; 2) T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 2 nghi\u1ec7m ph\u00e2n bi\u1ec7t; 3) T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 3 nghi\u1ec7m ph\u00e2n bi\u1ec7t. B\u00e0i 7. Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh: 4) x2 + 3x 2 + 3x2 + 9x + 2 = 0; 1) x4 \u2212 3x2 + 4 = 0; | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT 2) 2x2 + 1 2 \u2212 4 = 0; 5) 1 3 + 2x3 + 3 = 17 3) 3x2 + 5 2 + 15x2 + 19 = 0; 2x2 + . 4 B\u00e0i 8. Ph\u01b0\u01a1ng tr\u00ecnh (x \u2212 1) x2 + 5x \u2212 2m = 0. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 ba nghi\u1ec7m ph\u00e2n bi\u1ec7t. B\u00e0i 9. Cho x2 + 5mx + 2m \u2212 1 = 0. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 nghi\u1ec7m k\u00e9p. B\u00e0i 10. Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh: 5x2 + 16x + 11 = 0. B\u00e0i 11. Cho ph\u01b0\u01a1ng tr\u00ecnh mx2 + (4m + 1) x + 3m + 1 = 0. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m x1; x2 sao cho x12 + x22 = 1. B\u00e0i 12. Cho ph\u01b0\u01a1ng tr\u00ecnh (2m \u2212 1) x2 + (5m \u2212 2) x \u2212 7m + 3 = 0. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m x1; x2 sao cho x14 + x42 = 1. B\u00e0i 13. Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh: (x \u2212 1) x2 \u2212 5x + 6 = 0. B\u00e0i 14. Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh: x3 + 11x2 + 2x \u2212 18 = 0 B\u00e0i 15. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh 2mx2 = (5m \u2212 3) x \u2212 7m + 3 = 0 th\u1ecfa x13 + x32 = 1. B\u00e0i 16. Cho ph\u01b0\u01a1ng tr\u00ecnh (x \u2212 m) x2 + 2x + 3m \u2212 3 = 0. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 3 ngi\u1ec7m ph\u00e2n bi\u1ec7t. B\u00e0i 17. Cho ph\u01b0\u01a1ng tr\u00ecnh x3 \u2212 (5 + m) x2 + (4 + 5m) x \u2212 4m = 0. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t. B\u00e0i 18. Cho ph\u01b0\u01a1ng tr\u00ecnh (x \u2212 m) x2 + 5x + 6 = 0. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng t\u00ecnh c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t. B\u00e0i 19. Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh sau: x3 \u2212 5x2 + 6x \u2212 2 = 0. B\u00e0i 20. Cho x2 \u2212 5x + 2m \u2212 1 = 0. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m x1; x2 sao cho x1 + 3x2 = 4. B\u00e0i 21. Cho ph\u01b0\u01a1ng tr\u00ecnh (x \u2212 m) x2 + 5x + 6 = 0. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t. \u221a B\u00e0i 22. Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh sau: x \u2212 3 x + 4 = 0. \u221a B\u00e0i 23. Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh sau: 4x \u2212 2x + 5 + 6 = 0. 50 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 B\u00e0i 24. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh x2 \u2212 6x + 2m \u2212 1 = 0 c\u00f3 m\u1ed9t nghi\u1ec7m l\u00e0 2. T\u00ecm nghi\u1ec7m c\u00f2n l\u1ea1i. B\u00e0i 25. Cho ph\u01b0\u01a1ng tr\u00ecnh x2 \u2212 (m + 4) x + 3m + 3 = 0, (x l\u00e0 \u1ea9n s\u1ed1, m l\u00e0 tham s\u1ed1). T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 m\u1ed9t nghi\u1ec7m b\u1eb1ng 2. T\u00ecm nghi\u1ec7m c\u00f2n l\u1ea1i. B\u00e0i 26. Cho ph\u01b0\u01a1ng tr\u00ecnh x2 \u2212 2 (m \u2212 1) x + 3n = 0, (x l\u00e0 \u1ea9n, m, n l\u00e0 c\u00e1c tham s\u1ed1). T\u00ecm m, n \u221a \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m l\u00e0 \u22124 v\u00e0 2 3. B\u00e0i 27. Cho ph\u01b0\u01a1ng tr\u00ecnh x2 + mx + 2m \u2212 4 = 0. 1) Ch\u1ee9ng t\u1ecf ph\u01b0\u01a1ng tr\u00ecnh lu\u00f4n c\u00f3 hai nghi\u1ec7m v\u1edbi m\u1ecdi gi\u00e1 tr\u1ecb c\u1ee7a m; 2) T\u00ednh t\u1ed5ng v\u00e0 t\u00edch hai nghi\u1ec7m theo m; | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M 3) T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m x1, x2 th\u1ecfa: x12 + x22 = 4. B\u00e0i 28. Cho ph\u01b0\u01a1ng tr\u00ecnh x2 + (2m \u2212 1) x + 2m \u2212 2 = 0. 1) Ch\u1ee9ng t\u1ecf ph\u01b0\u01a1ng tr\u00ecnh lu\u00f4n c\u00f3 hai nghi\u1ec7m v\u1edbi m\u1ecdi gi\u00e1 tr\u1ecb c\u1ee7a m; 2) G\u1ecdi x1, x2 l\u00e0 hai nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh. T\u00ecm m \u0111\u1ec3: x12 + x22 = 1. B\u00e0i 29. Cho ph\u01b0\u01a1ng tr\u00ecnh: x2 \u2212 2x + m \u2212 3 = 0. 1) Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh v\u1edbi m = 3; 2) \u0110\u1ecbnh m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 nghi\u1ec7m k\u00e9p. T\u00ecm nghi\u1ec7m k\u00e9p \u0111\u00f3. B\u00e0i 30. Cho ph\u01b0\u01a1ng tr\u00ecnh x2 + (m \u2212 2) x \u2212 m + 1 = 0. 1) Ch\u1ee9ng t\u1ecf ph\u01b0\u01a1ng tr\u00ecnh lu\u00f4n c\u00f3 hai nghi\u1ec7m v\u1edbi m\u1ecdi gi\u00e1 tr\u1ecb c\u1ee7a m; 2) T\u00ednh t\u1ed5ng v\u00e0 t\u00edch hai nghi\u1ec7m theo m; 3) T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m x1, x2 th\u1ecfa: x21x2 + x1x22 \u2212 4x1x2 = \u22122. B\u00e0i 31. Cho ph\u01b0\u01a1ng tr\u00ecnh x2 + (2m \u2212 1) x \u2212 m = 0. 1) Ch\u1ee9ng t\u1ecf r\u1eb1ng ph\u01b0\u01a1ng tr\u00ecnh lu\u00f4n c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t v\u1edbi m\u1ecdi m; 2) T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m th\u1ecfa: (x1 + x2)2 \u2212 4x1x2 = 5. B\u00e0i 32. Cho ph\u01b0\u01a1ng tr\u00ecnh x2 + 2mx \u2212 2m2 = 0. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m x1, x2 th\u1ecfa: x1 + x2 = x1x2. B\u00e0i 33. V\u1edbi gi\u00e1 tr\u1ecb n\u00e0o c\u1ee7a m th\u00ec ph\u01b0\u01a1ng tr\u00ecnh x2 + (m \u2212 1) x + m \u2212 2 = 0, (xl\u00e0 \u1ea9n s\u1ed1, m l\u00e0 tham s\u1ed1) c\u00f3 nghi\u1ec7m ph\u00e2n bi\u1ec7t c\u00f9ng \u00e2m. B\u00e0i 34. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh x2 \u2212 (m \u2212 1) x \u2212 m = 0 c\u00f3 hai nghi\u1ec7m c\u00f9ng \u00e2m. B\u00e0i 35. Cho ph\u01b0\u01a1ng tr\u00ecnh mx2 \u2212 2 (m \u2212 2) x + m \u2212 3 = 0. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m th\u1ecfa h\u1ec7 th\u1ee9c: (2x1 + 1) (2x2 + 1) = 25. B\u00e0i 36. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh x2 \u2212 5x + m \u2212 2 = 0 c\u00f3 hai nghi\u1ec7m x1, x2 th\u1ecfa: 1 + 1 = 5 x1 x2 . 6 B\u00e0i 37. Ph\u01b0\u01a1ng tr\u00ecnh: x2 \u2212 2 (m \u2212 1) x + m2 \u2212 1 = 0. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m c\u00f9ng d\u1ea5u. | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 51","| C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT | Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m B\u00e0i 38. Ph\u01b0\u01a1ng tr\u00ecnh x2 \u2212 3x + m \u2212 2 = 0. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m tr\u00e1i d\u1ea5u. B\u00e0i 39. Cho ph\u01b0\u01a1ng tr\u00ecnh: x2 \u2212 2 (m \u2212 1) x + m2 \u2212 3m = 0. 1) T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m tr\u00e1i d\u1ea5u; 2) T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 \u0111\u00fang m\u1ed9t nghi\u1ec7m \u00e2m; 3) T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m x1; x2 th\u1ecfa m\u00e3n h\u1ec7 th\u1ee9c x12 + x22 = 4; 4) V\u1edbi gi\u00e1 tr\u1ecb n\u00e0o c\u1ee7a m, ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m tr\u00e1i d\u1ea5u v\u00e0 nghi\u1ec7m \u00e2m c\u00f3 gi\u00e1 tr\u1ecb tuy\u1ec7t \u0111\u1ed1i l\u1edbn h\u01a1n nghi\u1ec7m d\u01b0\u01a1ng. B\u00e0i 40. Cho ph\u01b0\u01a1ng tr\u00ecnh mx2 \u2212 2 (m \u2212 1) x + 3m + 5 = 0. 1) T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m tr\u00e1i d\u1ea5u; 2) V\u1edbi gi\u00e1 tr\u1ecb n\u00e0o c\u1ee7a m, ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m tr\u00e1i d\u1ea5u v\u00e0 nghi\u1ec7m \u00e2m c\u00f3 gi\u00e1 tr\u1ecb tuy\u1ec7t \u0111\u1ed1i l\u1edbn h\u01a1n nghi\u1ec7m d\u01b0\u01a1ng. B\u00e0i 41. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh x2 + (2m \u2212 1) x + m2 + 3 = 0 c\u00f3 nghi\u1ec7m k\u00e9p, t\u00ecm nghi\u1ec7m k\u00e9p \u0111\u00f3. B\u00e0i 42. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh x2 \u2212 (m \u2212 1) x \u2212 m = 0 c\u00f3 2 nghi\u1ec7m c\u00f9ng \u00e2m. B\u00e0i 43. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh 7x2 + 2 (m \u2212 1) x \u2212 m2 = 0 c\u00f3 nghi\u1ec7m. B\u00e0i 44. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh x2 \u2212 8x + m = 0 c\u00f3 hai nghi\u1ec7m th\u1ecfa: x12 + x22 = 5. B\u00e0i 45. Cho ph\u01b0\u01a1ng tr\u00ecnh x2 + x \u2212 m2 = 0. 1) T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t; 2) Ch\u1ee9ng minh r\u1eb1ng x31 + x23 = \u22123m2 \u2212 1; 3) T\u00ecm m \u0111\u1ec3 A = x31 + x23 \u0111\u1ea1t gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t. B\u00e0i 46. Cho ph\u01b0\u01a1ng tr\u00ecnh x2 + 2mx + 2m \u2212 1 = 0. 1) Ch\u1ee9ng minh r\u1eb1ng ph\u01b0\u01a1ng tr\u00ecnh lu\u00f4n c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t v\u1edbi m\u1ecdi m; 2) T\u00ecm gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a A = x21 + x22 \u2212 4x1x2. B\u00e0i 47. T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh x2 \u2212 mx + 2 = 0 c\u00f3 m\u1ed9t nghi\u1ec7m b\u1eb1ng 3, t\u00ecm nghi\u1ec7m c\u00f2n l\u1ea1i. B\u00e0i 48. Ph\u01b0\u01a1ng tr\u00ecnh x2 \u2212 7x + m = 0 c\u00f3 m\u1ed9t nghi\u1ec7m b\u1eb1ng 3 th\u00ec m b\u1eb1ng bao nhi\u00eau? T\u00ecm nghi\u1ec7m c\u00f2n l\u1ea1i. B\u00e0i 49. Cho ph\u01b0\u01a1ng tr\u00ecnh 2x2 \u2212 4mx + 2m2 \u2212 m \u2212 3 = 0. 1) T\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t x1, x2; 2) T\u00ecm gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a A = 5x1 + 5x2 \u2212 4x1x2. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 52 N\u0103m h\u1ecdc: 2023 - 2024","Ch\u01b0\u01a1ng 1 H\u1ec6 TH\u1ee8C L\u01af\u1ee2NG TRONG TAM GI\u00c1C VU\u00d4NG B\u00e0i 1 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M H\u1ec7 th\u1ee9c l\u01b0\u1ee3ng trong tam gi\u00e1c vu\u00f4ng Cao I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: Trong tam gi\u00e1c vu\u00f4ng c\u00f3 \u0111\u01b0\u1eddng cao, ta c\u00f3 c\u00e1c h\u1ec7 th\u1ee9c sau: 1) Vu\u00f4ng2 = Chi\u1ebfu \u00d7 Huy\u1ec1n A 2) Huy\u1ec3n2 = Vu\u00f4ng2 + Vu\u00f4ng2 Vu\u00f4ng 3) Cao \u00d7 Huy\u1ec1n = Vu\u00f4ng \u00d7 Vu\u00f4ng Vu\u00f4ng 4) Cao2 = Chi\u1ebfu \u00d7 Chi\u1ebfu Chi\u1ebfu Chi\u1ebfu BH Huy\u1ec1n 11 1 C 5) Cao2 = Vu\u00f4ng2 + Vu\u00f4ng2 II. B\u00e0i t\u1eadp: 3cm 6cm A 8cm B\u00e0i 1. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH. Bi\u1ebft AB = 6cm, AC = 8cm. L\u1ea7n l\u01b0\u1ee3t t\u00ednh \u0111\u1ed9 d\u00e0i c\u00e1c \u0111o\u1ea1n th\u1eb3ng: BC, AH, BH, CH. B H C A B\u00e0i 2. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH. Bi\u1ebft AB = 3cm, BC = 5cm. L\u1ea7n l\u01b0\u1ee3t t\u00ednh \u0111\u1ed9 d\u00e0i c\u00e1c \u0111o\u1ea1n th\u1eb3ng: BH, CH, AH, AC. B H 5cm C B\u00e0i 3. Cho \u25b3M N P vu\u00f4ng t\u1ea1i M c\u00f3 \u0111\u01b0\u1eddng cao M H. Bi\u1ebft M P = 10cm, P H = 5cm. L\u1ea7n l\u01b0\u1ee3t | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 53","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m t\u00ednh \u0111\u1ed9 d\u00e0i c\u00e1c \u0111o\u1ea1n th\u1eb3ng sau: P N, HN, M H, M N . I B\u00e0i 4. T\u00ecm x, y trong c\u00e1c tam gi\u00e1c sau: y M 8cm | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT 6cm 2cm x y 1cm x PD N JM K B\u00e0i 5. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A, \u0111\u01b0\u1eddng cao AH. Cho CH = 2cm, BH = 8cm. T\u00ednh AH. B\u00e0i 6. Cho \u25b3DEF vu\u00f4ng t\u1ea1i D c\u00f3 DH l\u00e0 \u0111\u01b0\u1eddng cao. Bi\u1ebft ED = 6cm, DH = 4, 8cm. L\u1ea7n l\u01b0\u1ee3t t\u00ednh \u0111\u1ed9 d\u00e0i c\u00e1c \u0111o\u1ea1n th\u1eb3ng sau: DF, EF, HE, HF . B\u00e0i 7. Cho \u25b3DEF vu\u00f4ng t\u1ea1i D c\u00f3 DH l\u00e0 \u0111\u01b0\u1eddng cao. Bi\u1ebft DH = 5cm, HF = 12cm. L\u1ea7n l\u01b0\u1ee3t t\u00ednh \u0111\u1ed9 d\u00e0i c\u00e1c \u0111o\u1ea1n th\u1eb3ng: DF, EH, EF, DE. B\u00e0i 8. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A, \u0111\u01b0\u1eddng cao AH. Bi\u1ebft CH = 4, 5cm, BH = 8cm. T\u00ednh AH, AB, AC . B\u00e0i 9. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A, \u0111\u01b0\u1eddng cao AH. Bi\u1ebft CH = 9cm, AH = 12cm. T\u00ednh BC, AB, AC . B\u00e0i 10. Cho \u25b3M N P vu\u00f4ng t\u1ea1i M , \u0111\u01b0\u1eddng cao M H. Bi\u1ebft M P = 15cm, M H = 12cm. T\u00ednh chu vi v\u00e0 di\u1ec7n t\u00edch c\u1ee7a tam gi\u00e1c M N P . B\u00e0i 11. Cho \u25b3ABC bi\u1ebft AB = 8cm, AC = 15cm, BC = 17cm. Ch\u1ee9ng minh r\u1eb1ng \u25b3ABC l\u00e0 tam gi\u00e1c vu\u00f4ng. B\u00e0i 12. Cho \u25b3ABC c\u00f3 \u0111\u01b0\u1eddng cao AH. T\u00ednh AH bi\u1ebft, AB = 5cm, AC = 12cm v\u00e0 BC = 13cm. B\u00e0i 13. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH. Bi\u1ebft AB = 6cm, CH = 6, 4cm. GiBH = x. 1) Ch\u1ee9ng minh r\u1eb1ng x(x + 6, 4) = 36; 2) T\u00ecm x r\u1ed3i t\u00ednh BC, AH, AC. B\u00e0i 14. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH. Bi\u1ebft AB = 6cm, CH = 9cm. T\u00ednh: BH, BC, AH, AC. B\u00e0i 15. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH. Bi\u1ebft AH = 4, 8cm, BC = 10cm v\u00e0 AB < AC. T\u00ednh: AB, AC. B\u00e0i 16. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH. Bi\u1ebft AH = 6, 72cm, BC = 25cm v\u00e0 AB < AC. T\u00ednh: AB, AC. B\u00e0i 17. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH. Bi\u1ebft AB = 6cm, CH = 9cm. T\u00ednh: BH, BC, AH, AC. B\u00e0i 18. Cho \u25b3ABC c\u00f3 \u0111\u01b0\u1eddng cao AH. Ch\u1ee9ng minh r\u1eb1ng \u25b3ABC vu\u00f4ng n\u1ebfu bi\u1ebft AH = 12cm, 54 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9SS CH = 16cm, BH = 9cm | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M B\u00e0i 19. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH. Ch\u1ee9ng minh r\u1eb1ng: 1) \u25b3ABH \u25b3CBA; 2) BA2 = BH.BC. B\u00e0i 20. Cho tam gi\u00e1c M N P vu\u00f4ng t\u1ea1i M c\u00f3 \u0111\u01b0\u1eddng cao M H. Ch\u1ee9ng minh r\u1eb1ng: 1) \u25b3M N H \u25b3P M H; 2) M H2 = N H.HP . B\u00e0i 21. Cho \u25b3DF E vu\u00f4ng t\u1ea1i D c\u00f3 \u0111\u01b0\u1eddng cao DK. Ch\u1ee9ng minh r\u1eb1ng: 1) DF 2 = F K.F E; 2) DF.DE = DK.EF . B\u00e0i 22. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH. K\u1ebb HE vu\u00f4ng g\u00f3c v\u1edbi AB t\u1ea1i E, HF vu\u00f4ng g\u00f3c v\u1edbi AC t\u1ea1i F . Ch\u1ee9ng minh r\u1eb1ng: 1) AH2 = AB.AE; 2) AH2 = AF.AC; 3) AF.AC = AB.AE. B\u00e0i 23. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH. K\u1ebb HE vu\u00f4ng g\u00f3c v\u1edbi AB t\u1ea1i E, HF vu\u00f4ng g\u00f3c v\u1edbi AC t\u1ea1i F . Ch\u1ee9ng minh r\u1eb1ng \u25b3AF E \u25b3ABC. B\u00e0i 24. Cho \u25b3ABC nh\u1ecdn c\u00f3 \u0111\u01b0\u1eddng cao AD, v\u1ebd DE \u22a5 AB t\u1ea1i E, DF \u22a5 AC t\u1ea1i F . Ch\u1ee9ng minh r\u1eb1ng \u25b3AEF \u25b3ACB. B\u00e0i 25. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH. G\u1ecdi E v\u00e0 F l\u1ea7n l\u01b0\u1ee3t l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a H l\u00ean AB v\u00e0 AC. Ch\u1ee9ng minh r\u1eb1ng: 1) AH3 = BE.CF.BC; 2) BH3 = BE2.BC. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 S S B\u00e0i 2 T\u1ef7 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: 1. \u0110\u1ecbnh ngh\u0129a: Trong tam gi\u00e1c vu\u00f4ng, c\u00f3 g\u00f3c nh\u1ecdn \u03b1, ta c\u00f3 c\u00e1c t\u1ef7 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c nh\u01b0 sau: | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 55","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m \u0110\u1ed1i \u0110\u1ed1i c\u1ea1nh huy\u1ec1n sin \u03b1 = tan \u03b1 = \u03b1 Huy\u1ec1n K\u1ec1 c\u1ea1nh k\u1ec1 K\u1ec1 K\u1ec1 cot \u03b1 = cos \u03b1 = \u0110\u1ed1i Huy\u1ec1n | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT c\u1ea1nh \u0111\u1ed1i2. T\u00ednh ch\u1ea5t: C\u00e1c t\u1ef7 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c sin, cos lu\u00f4n l\u1edbn h\u01a1n 0 v\u00e0 b\u00e9 h\u01a1n 1. N\u1ebfu hai g\u00f3c ph\u1ee5 nhau th\u00ec sin g\u00f3c n\u00e0y b\u1eb1ng cos g\u00f3c kia, tan g\u00f3c n\u00e0y b\u1eb1ng cot g\u00f3c kia v\u00e0 ng\u01b0\u1ee3c l\u1ea1i. Khi g\u00f3c t\u0103ng th\u00ec sin v\u00e0 tan t\u0103ng, cos v\u00e0 cot gi\u1ea3m. II. B\u00e0i t\u1eadp: B\u00e0i 1. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A. T\u00ednh c\u00e1c t\u1ec9 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c c\u1ee7a g\u00f3c B v\u00e0 g\u00f3c C n\u1ebfu bi\u1ebft: 1) AB = 4cm; AC = 3cm; 2) AB = 6cm; AC = 10cm; 3) AB = 5cm; AC = 12cm; 4) AB = 4cm; AC = 3cm. B\u00e0i 2. Cho tan gi\u00e1c ABC vu\u00f4ng t\u1ea1i A, AH l\u00e0 \u0111\u01b0\u1eddng cao. T\u00ednh t\u1ec9 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c c\u1ee7a B r\u1ed3i suy ra t\u1ec9 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c c\u1ee7a C: 1) AB = 30cm, AH = 24cm; 2) BH = 9dm, AH = 12dm; 3) AB = 6cm, BH = 3, 6cm. B\u00e0i 3. T\u00ecm g\u00f3c x bi\u1ebft sin x = 0, 26; cos x = 0, 3; tan x = 2, 6; cot x = 4. B\u00e0i 4. S\u1eed d\u1ee5ng m\u00e1y t\u00ednh h\u00e3y t\u00ednh gi\u00e1 tr\u1ecb c\u00e1c t\u1ec9 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c c\u1ee7a c\u00e1c g\u00f3c \u0111\u1eb7t bi\u1ec7t sau: G\u00f3c x 15\u25e6 30\u25e6 45\u25e6 60\u25e6 75\u25e6 sin x cos x tan x cot x D\u01b0a v\u00e0o c\u00e1c gi\u00e1 tr\u1ecb lu\u1ecdng gi\u00e1c trong b\u1ea3ng tr\u00ean, h\u00e3y nh\u1eadn x\u00e9t s\u1ef1 t\u0103ng gi\u1ea3m c\u1ee7a t\u00f9ng lo\u1ea1i gi\u00e1 tr\u1ecb lu\u1ee3ng gi\u00e1c ph\u1ee5 thu\u1ed9c nh\u01b0 th\u1ebf n\u00e0o v\u00e0o s\u1ef1 t\u0103ng gi\u1ea3m c\u1ee7a c\u00e1c g\u00f3c t\u01b0\u01a1ng \u1ee9ng. B\u00e0i 5. Gi\u1ea3i tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A n\u1ebfu bi\u1ebft: 1) A = B = 6cm, AC = 8cm; 56 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M 2) AB = 5cm, BC = 13cm; 1 3) AB = 12cm, cos B = ; 2 \u221a1 4) AB = 3cm, sin B = ; \u221a2 5) AB = 4cm, tan B = 3; \u221a 6) BC = 2cm, cot B = 1; \u221a 7) AC = 5 3cm, C\u02c6 = 30\u25e6. \u221a B\u00e0i 6. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A, \u0111\u01b0\u1eddng cao AH. Bi\u1ebft BH = 1cm, AH = 3cm. T\u00ednh ACB. B\u00e0i 7. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A, \u0111\u01b0\u1eddng cao AH = 5cm, B = 60\u25e6. T\u00ednh AC. B\u00e0i 8. Gi\u1ea3i \u25b3ABC vu\u00f4ng t\u1ea1i A, bi\u1ebft: 1) AB = 5cm, AC = 12cm; 2) AC = 5cm, B = 40\u25e6. B\u00e0i 9. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A, B = 60\u25e6, \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng cao AH = 4cm. T\u00ednh AC. B\u00e0i 10. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A, AB = 6cm, AC = 12cm. T\u00ednh sin B, cos B, tan C. \u221a B\u00e0i 11. Cho \u25b3ABC c\u00f3 B = 30\u25e6, C = 45\u25e6, AC = 3 2. T\u00ednh AB. B\u00e0i 12. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A, AB = 14cm, BC = 50cm. T\u00ednh sin B + cos C. B\u00e0i 13. S\u1eafp x\u1ebfp c\u00e1c t\u1ec9 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c sau: 1) Theo th\u1ee9 t\u1ef1 gi\u1ea3m d\u1ea7n: sin 25\u25e6, cos 80\u25e6, sin 16\u25e6, cos 70\u25e6, sin 55\u25e6, cos 50\u25e6; 2) Theo th\u1ee9 t\u1ef1 gi\u1ea3m d\u1ea7n: tan 25\u25e6, cot 70\u25e6, tan 26\u25e6, cot 40\u25e6, tan 15\u25e6, cot 50\u25e6; 3) Theo th\u1ee9 t\u1ef1 t\u0103ng d\u1ea7n: sin 32\u25e6, cos 32\u25e6, sin 50\u25e6, cos 73\u25e6; 4) Theo th\u1ee9 t\u1ef1 gi\u1ea3m d\u1ea7n: sin 35\u25e6, cot 17\u25e6, tan 83\u25e6, cot 65\u25e6; 5) Theo th\u1ee9 t\u1ef1 gi\u1ea3m d\u1ea7n: tan 42\u25e6, cot 35\u25e6, tan 67\u25e6, cot 83\u25e6; 6) Theo th\u1ee9 t\u1ef1 t\u0103ng d\u1ea7n: sin 35\u25e6; cos 67\u25e6; sin 80\u25e6; cos 35\u25e6; 7) Theo th\u1ee9 t\u1ef1 t\u0103ng d\u1ea7n: sin 27\u25e6; cos 31\u25e6; sin 45\u25e6; cos 70\u25e6; 8) T\u1eeb nh\u1ecf \u0111\u1ebfn l\u1edbn: cos 48\u25e6; sin 48\u25e6; sin 35\u25e6; sin 75\u25e6; cos 62\u25e6; 9) S\u1eafp x\u1ebfp theo th\u1ee9 t\u1ef1 t\u0103ng d\u1ea7n: sin 15\u25e6; cos 80\u25e6; tan 25\u25e6; cot 75\u25e6; 10) S\u1eafp x\u1ebfp theo th\u1ee9 t\u1ef1 t\u0103ng d\u1ea7n: sin 10\u25e6; cos 10\u25e6; tan 45\u25e6; cos 33\u25e6. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 57","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m B\u00e0i 3 B\u00e0i to\u00e1n th\u1ef1c t\u1ebf | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT B\u00e0i 1. M\u1ed9t c\u1ed9t \u0111\u00e8n cao 7m c\u00f3 b\u00f3ng tr\u00ean m\u1eb7t \u0111\u1ea5t d\u00e0i 5m. Khi \u0111\u00f3 tia s\u00e1ng m\u1eb7t tr\u1eddi t\u1ea1o v\u1edbi m\u1eb7t \u0111\u1ea5t m\u1ed9t g\u00f3c bao nhi\u00eau \u0111\u1ed9. (l\u00e0m tr\u00f2n \u0111\u1ebfn \u0111\u1ed9). B\u00e0i 2. M\u1ed9t c\u00e1i th\u00e1p truy\u1ec1n h\u00ecnh c\u00f3 b\u00f3ng ng\u00e3 tr\u00ean m\u1eb7t \u0111\u1ea5t d\u00e0i 58m. Bi\u1ebft c\u00e1c tia n\u1eafng m\u1eb7t tr\u1eddi t\u1ea1o v\u1edbi m\u1eb7t \u0111\u1ea5t m\u1ed9t g\u00f3c 38\u25e6. T\u00ednh chi\u1ec1u cao c\u1ee7a c\u00e1i th\u00e1p \u0111\u00f3. (L\u00e0m tr\u00f2n \u0111\u1ebfn ch\u1eef s\u1ed1 th\u1eadp ph\u00e2n th\u1ee9 nh\u1ea5t). B\u00e0i 3. M\u1ed9t c\u00e2y c\u1ed9t \u0111i\u1ec7n cao 12m c\u00f3 b\u00f3ng tr\u00ean m\u1eb7t \u0111\u1ea5t d\u00e0i 8m. Khi \u0111\u00f3 tia s\u00e1ng m\u1eb7t tr\u1eddi t\u1ea1o v\u1edbi m\u1eb7t \u0111\u1ea5t m\u1ed9t g\u00f3c bao nhi\u00eau \u0111\u1ed9? B\u00e0i 4. M\u1ed9t c\u00e1i thang d\u00e0i 3m d\u1ef1a v\u00e0o t\u01b0\u1eddng t\u1ea1o th\u00e0nh g\u00f3c 60\u25e6 v\u1edbi m\u1eb7t \u0111\u1ea5t. H\u1ecfi ch\u00e2n thang c\u00e1ch t\u01b0\u1eddng bao nhi\u00eau m\u00e9t? B\u00e0i 5. M\u1ed9t c\u00e2y th\u00f4ng cao 8 m c\u00f3 b\u00f3ng tr\u00ean m\u1eb7t \u0111\u1ea5t d\u00e0i 6m. Khi \u0111\u00f3 tia s\u00e1ng m\u1eb7t tr\u1eddi t\u1ea1o v\u1edbi m\u1eb7t \u0111\u1ea5t m\u1ed9t g\u00f3c bao nhi\u00eau \u0111\u1ed9. (l\u00e0m tr\u00f2n \u0111\u1ebfn \u0111\u1ed9). B\u00e0i 6. M\u1ed9t c\u00e1i th\u00e1p truy\u1ec1n h\u00ecnh c\u00f3 b\u00f3ng ng\u00e3 tr\u00ean m\u1eb7t \u0111\u1ea5t d\u00e0i 55m. Bi\u1ebft c\u00e1c tia n\u1eafng m\u1eb7t tr\u1eddi t\u1ea1o v\u1edbi m\u1eb7t \u0111\u1ea5t m\u1ed9t g\u00f3c 42\u25e6. T\u00ednh chi\u1ec1u cao c\u1ee7a c\u00e1i th\u00e1p \u0111\u00f3. (L\u00e0m tr\u00f2n \u0111\u1ebfn ch\u1eef s\u1ed1 th\u1eadp ph\u00e2n th\u1ee9 hai). B\u00e0i 7. M\u1ed9t con m\u00e8o \u1edf tr\u00ean c\u00e0nh c\u00e2y cao 6,5m. \u0110\u1ec3 b\u1eaft con m\u00e8o xu\u1ed1ng c\u1ea7n ph\u1ea3i \u0111\u1eb7t thang sao cho \u0111\u1ea7u thang \u0111\u1ea1t \u0111\u1ed9 cao \u0111\u00f3. Khi \u0111\u00f3 g\u00f3c c\u1ee7a thang v\u1edbi m\u1eb7t \u0111\u1ea5t l\u00e0 bao nhi\u00eau, bi\u1ebft r\u1eb1ng c\u1ea7u thang d\u00e0i 6, 7 m ? \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 58 N\u0103m h\u1ecdc: 2023 - 2024","B\u00e0i 4 GI\u00c1O TR\u00ccNH TO\u00c1N 9 C\u00e1c h\u1ec7 th\u1ee9c l\u01b0\u1ee3ng gi\u00e1c c\u01a1 b\u1ea3n I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: 1) sin2 \u03b1 + cos2 \u03b1 = 1 sin \u03b1 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M 2) tan \u03b1 = cos \u03b1 cos \u03b1 3) cot \u03b1 = sin \u03b1 4) tan \u03b1 \u00d7 cot \u03b1 = 1 5) 1 + tan2 \u03b1 = 1 cos2 \u03b1 6) 1 + cot2 \u03b1 = 1 sin2 \u03b1 II. B\u00e0i t\u1eadp: B\u00e0i 1. \u0110\u01a1n gi\u1ea3n bi\u1ec3u th\u1ee9c: 1) M = 1 + sin2 x + cos2 x; 2) N = sin x \u2212 sin x. cos2 x. B\u00e0i 2. \u0110\u01a1n gi\u1ea3n bi\u1ec3u th\u1ee9c: 1) 1 \u2212 sin2 \u03b1; 2) (1 \u2212 cos \u03b1)(1 + cos \u03b1). B\u00e0i 3. \u0110\u01a1n gi\u1ea3n c\u00e1c bi\u1ec3u th\u1ee9c sau: 1) 1 + sin2 \u03b1 + cos2 \u03b1; 2) cos \u03b1 \u2212 cos \u03b1 \u00b7 sin2 \u03b1. B\u00e0i 4. Ch\u1ee9ng minh r\u1eb1ng: 1) 1 + tan2 \u03b1 = 1 cos2 \u03b1 ; 2) 1 + cot2 \u03b1 = 1 sin2 \u03b1 . B\u00e0i 5. Ch\u1ee9ng minh r\u1eb1ng sin4 \u03b1 + cos4 \u03b1 + 2 sin2 \u03b1. cos2 \u03b1 = 1. B\u00e0i 6. \u0110\u01a1n gi\u1ea3n bi\u1ec3u th\u1ee9c B = sin \u03b1 \u2212 sin \u03b1. cos2 \u03b1. B\u00e0i 7. R\u00fat g\u1ecdn A = tan2 \u03b1 2 cos2 \u03b1 + sin2 \u03b1 \u2212 1 . B\u00e0i 8. Ch\u1ee9ng minh r\u1eb1ng cos2 \u03b1 + tan2 \u03b1. cos2 \u03b1 = 1. B\u00e0i 9. R\u00fat g\u1ecdn: P = cot2 \u03b1 2 sin2 \u03b1 + cos2 \u03b1 \u2212 1 . | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 59","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m B\u00e0i 10. Ch\u1ee9ng minh r\u1eb1ng tan2 \u03b1 \u2212 sin2 \u03b1. tan2 \u03b1 = sin2 \u03b1. B\u00e0i 11. R\u00fat g\u1ecdn A = tan2 \u03b1 2 sin2 \u03b1 + cos2 \u03b1 \u2212 1 B\u00e0i 12. Cho \u03b1 l\u00e0 g\u00f3c nh\u1ecdn. Ch\u1ee9ng minh r\u1eb1ng tan2 \u03b1 \u2212 sin2 \u03b1 = tan2 \u03b1. sin2 \u03b1. B\u00e0i 13. R\u00fat g\u1ecdn bi\u1ec3u th\u1ee9c: 1) A = 2 cos2 \u03b1 \u2212 1 ; sin \u03b1 + cos \u03b1 2) cos2 \u03b1 \u00b7 cot2 \u03b1 \u2212 cot2 \u03b1. | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT B\u00e0i 14. T\u00ednh gi\u00e1 tr\u1ecb c\u00e1c bi\u1ec3u th\u1ee9c sau: 1) sin 23\u25e6 \u2212 cos 67\u25e6; 2) cos 34\u25e6 \u2212 sin 56\u25e6; 3) tan 15\u25e6. tan 35\u25e6. tan 55\u25e6. tan 75\u25e6; 4) cot 36\u25e6 \u2212 tan 54\u25e6; 5) tan 18\u25e6 \u2212 cot 72\u25e6; 6) cot 30\u25e6. cot 20\u25e6. cot 60\u25e6. cot 70\u25e6. B\u00e0i 15. T\u00ednh gi\u00e1 tr\u1ecb c\u00e1c bi\u1ec3u th\u1ee9c sau: tan 27\u25e6. tan 63\u25e6 1) A = cot 63\u25e6. cot 27\u25e6 ; cot 20\u25e6. cot 45\u25e6. cot 70\u25e6 2) B = tan 20\u25e6. tan 45\u25e6. tan 70\u25e6 . B\u00e0i 16. T\u00ednh: A = sin2 36\u25e6 + sin2 54\u25e6 \u2212 tan 25\u25e6. tan 65\u25e6. B\u00e0i 17. T\u00ednh gi\u00e1 tr\u1ecb c\u00e1c bi\u1ec3u th\u1ee9c sau: 1) A = sin2 22\u25e6 + cos2 22\u25e6; 2) B = sin2 40\u25e6 + sin2 50\u25e6; 3) C = cos2 20\u25e6 + sin2 70\u25e6; 4) D = tan 15\u25e6. cot 15\u25e6; 5) E = tan 18\u25e6. tan 72\u25e6; 6) F = cot 16\u25e6. cot 74\u25e6. B\u00e0i 18. Ch\u1ee9ng minh r\u1eb1ng sin2 33\u25e6 + sin2 57\u25e6 + tan 28\u25e6. tan 62\u25e6 = 2. B\u00e0i 19. Kh\u00f4ng d\u00f9ng m\u00e1y t\u00ednh b\u1ecf t\u00fai, t\u00ednh gi\u00e1 tr\u1ecb bi\u1ec3u th\u1ee9c sau: M = tan 67\u25e6 \u2212 cot 23\u25e6 + cos2 16\u25e6 + cos2 74\u25e6 \u2212 cot 37\u25e6 tan 53\u25e6 . B\u00e0i 20. T\u00ednh: P = sin2 72\u25e6 + cos2 72\u25e6 + 2 tan 55\u25e6 cot 35\u25e6 . cot 32\u25e6 B\u00e0i 21. T\u00ednh: A = sin2 25\u25e6 + sin2 65\u25e6 \u2212 tan 35\u25e6 + cot 55\u25e6 \u2212 tan 58\u25e6 . 60 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 B\u00e0i 22. T\u00ednh: A = 2 cot 37\u25e6. cot 53\u25e6 + sin2 28\u25e6 \u2212 3 tan 54\u25e6 + sin2 62\u25e6. cot 36\u25e6 B\u00e0i 23. T\u00ednh: sin2 12\u25e6 + sin2 70\u25e6 \u2212 sin2 35\u25e6 + sin2 30\u25e6 + sin2 78\u25e6 \u2212 sin2 55\u25e6 + sin2 20\u25e6. B\u00e0i 24. Cho x l\u00e0 g\u00f3c nh\u1ecdn: 3 1) Bi\u1ebft sin x = . T\u00ednh cos x, tan x, cot x; 5 12 2) Bi\u1ebft cos x = . T\u00ednh sin x, tan x, cot x; 1\u221a3 3) Bi\u1ebft tan x = 3. T\u00ednh sin x, cos x, cot x; 4) Bi\u1ebft cot x = 1. T\u00ednh sin x, cos x, tan x. | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M 4 B\u00e0i 25. Bi\u1ebft sin \u03b1 = . T\u00ednh cos \u03b1, tan \u03b1, cot \u03b1. 5 B\u00e0i 26. T\u00ednh gi\u00e1 tr\u1ecb c\u00e1c bi\u1ec3u th\u1ee9c sau: 1) A = sin 10\u25e6 + sin 40\u25e6 \u2212 cos 50\u25e6 \u2212 cos 80\u25e6. 2) B = cos 15\u25e6 \u2212 sin 75\u25e6 + cos 35\u25e6 \u2212 sin 55\u25e6. B\u00e0i 27. Bi\u1ebft sin \u03b1 = 1 T\u00ednh 4 cos2 \u03b1 \u2212 6 sin2 \u03b1. . 5 B\u00e0i 28. T\u00ednh: cot \u03b1 bi\u1ebft sin \u03b1 = 0, 6. B\u00e0i 29. Bi\u1ebft tan \u03b1 = \u221a2. T\u00ednh cot \u03b1. 3 B\u00e0i 30. Bi\u1ebft sin \u03b1 = . Kh\u00f4ng t\u00ednh s\u1ed1 \u0111o g\u00f3c \u03b1, h\u00e3y t\u00ednh: cos \u03b1, tan \u03b1, cot \u03b1. 2 B\u00e0i 31. tan x = 3. A = sin3 x \u2212 cos3 x Cho H\u00e3y t\u00ednh: . sin3 x + cos3 x B\u00e0i 32. Cho 0\u25e6 < \u03b1 < 90\u25e6 v\u00e0 sin \u03b1 = 5 T\u00ednh A = cot2 \u03b1 2 sin2 \u03b1 + cos2 \u03b1 \u2212 1 . . 6 \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 5 \u00d4n t\u1eadp ch\u01b0\u01a1ng 1 B\u00e0i 1. H\u00e3y v\u1ebd tam gi\u00e1c DEF vu\u00f4ng t\u1ea1i D, \u0111\u01b0\u1eddng cao DH. Vi\u1ebft c\u00e1c h\u1ec7 th\u1ee9c v\u1ec1 c\u1ea1nh v\u00e0 \u0111\u01b0\u1eddng cao trong tam gi\u00e1c tr\u00ean. B\u00e0i 2. T\u00ecm x, bi\u1ebft BC = 10cm. A x B 60\u25e6 10cm C B\u00e0i 3. | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 61","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m N 1) T\u00ecm x tr\u00ean h\u00ecnh v\u1ebd sau: 2) Cho B = 50\u25e6, AC = 5cm. T\u00ednh AB. 4cm x 6cm x 9cm P 3) T\u00ecm x, y tr\u00ean h\u00ecnh v\u1ebd: MD 5cm C A C | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT B 50\u25e6 yy A x 3cm x BH B\u00e0i 4. Cho \u25b3M N P vu\u00f4ng t\u1ea1i P, bi\u1ebft MP = 8cm; tan M = 3 T\u00ednh PN. . 4 B\u00e0i 5. T\u00ecm x, y trong h\u00ecnh v\u1ebd b\u00ean: A B\u00e0i 6. T\u00ecm x, y tr\u00ean h\u00ecnh v\u1ebd b\u00ean: 7cm 12cm C BH C 10cm A 8cm B x H B\u00e0i 7. Cho h\u00ecnh v\u1ebd v\u1edbi AH = 3cm; HC = 10cm. T\u00ednh AB, A BH, BC. 3cm 10cm BH C B\u00e0i 8. Cho \u25b3ABC vu\u00f4ng t\u1ea1i C c\u00f3 CH l\u00e0 \u0111\u01b0\u1eddng cao bi\u1ebft HA = 25cm, HB = 4cm. T\u00ednh CH v\u00e0 AC . 62 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M B\u00e0i 9. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A, \u0111\u01b0\u1eddng cao AH. Bi\u1ebft AB = 12cm; AC = 5cm. T\u00ednh BC, AH, BH, CH. \u221a B\u00e0i 10. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i C c\u00f3 CH l\u00e0 \u0111\u01b0\u1eddng cao. Bi\u1ebft CH = 10cm, AC = 5 14cm. T\u00ednh BC, BAC (s\u1ed1 \u0111o g\u00f3c l\u00e0m tr\u00f2n \u0111\u1ebfn \u0111\u1ed9, \u0111\u1ed9 d\u00e0i l\u00e0m tr\u00f2n \u0111\u1ebfn ch\u1eef s\u1ed1 th\u1eadp ph\u00e2n th\u1ee9 nh\u1ea5t). B\u00e0i 11. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A \u0111\u01b0\u1eddng cao AH. Bi\u1ebft BH = 5cm, CH = 33, 8cm. T\u00ednh \u0111\u1ed9 d\u00e0i AH. B\u00e0i 12. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A, \u0111\u01b0\u1eddng cao AH. Bi\u1ebft AH = 12cm, BC = 25cm. T\u00ednh chu vi v\u00e0 di\u1ec7n t\u00edch tam gi\u00e1c ABC. B\u00e0i 13. Gi\u1ea3i tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A bi\u1ebft A = 17cm, C = 30\u25e6. B\u00e0i 14. Cho tam gi\u00e1c ABC c\u00f3 AB = 6cm; AC = 10cm; BC = 8cm; \u0111\u01b0\u1eddng cao BH. Ch\u1ee9ng minh r\u1eb1ng tam gi\u00e1c ABC vu\u00f4ng v\u00e0 t\u00ednh AH, g\u00f3c C. (s\u1ed1 \u0111o g\u00f3c l\u00e0m tr\u00f2n \u0111\u1ebfn ph\u00fat, \u0111\u1ed9 d\u00e0i l\u00e0m tr\u00f2n \u0111\u1ebfn ch\u1eef s\u1ed1 th\u1eadp ph\u00e2n th\u1ee9 nh\u1ea5t). B\u00e0i 15. Cho tam gi\u00e1c M N P c\u00f3 M H l\u00e0 \u0111\u01b0\u1eddng cao, bi\u1ebft N = 70\u25e6; P = 30\u25e6, HN = 5cm. T\u00ednh MH, MP. B\u00e0i 16. H\u00e3y v\u1ebd tam gi\u00e1c MNP vu\u00f4ng t\u1ea1i M , \u0111\u01b0\u1eddng cao M H. Vi\u1ebft c\u00e1c h\u1ec7 th\u1ee9c v\u1ec1 c\u1ea1nh v\u00e0 \u0111\u01b0\u1eddng cao trong tam gi\u00e1c MNP vu\u00f4ng t\u1ea1i M . B\u00e0i 17. Cho tam gi\u00e1c DEF vu\u00f4ng t\u1ea1i D \u0111\u01b0\u1eddng cao DH c\u00f3 DE = 12cm; DF = 16cm. T\u00ednh \u0111\u1ed9 d\u00e0i EF ; DH; HE; HF. B\u00e0i 18. Cho tam gi\u00e1c ABC, \u0111\u01b0\u1eddng cao AH. Bi\u1ebft B = 43\u25e6, C = 35\u25e6, c\u1ea1nh AB c\u00f3 \u0111\u1ed9 d\u00e0i b\u1eb1ng 24cm. T\u00ednh AH, AC, HC. (L\u00e0m tr\u00f2n \u0111\u1ebfn ch\u1eef s\u1ed1 th\u1eadp ph\u00e2n th\u1ee9 nh\u1ea5t). B\u00e0i 19. Cho tam gi\u00e1c M N P vu\u00f4ng t\u1ea1i M , \u0111\u01b0\u1eddng cao M I, M N = 5cm; M P = 20cm. T\u00ednh \u0111\u1ed9 d\u00e0i c\u00e1c \u0111o\u1ea1n th\u1eb3ng N P, M I, IN, IP. B\u00e0i 20. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A, \u0111\u01b0\u1eddng cao AH. Bi\u1ebft AB = 8cm; BH = 4cm. T\u00ednh BC, HC, AH (K\u1ebft qu\u1ea3 l\u00e0m tr\u00f2n \u0111\u1ebfn ch\u1eef s\u1ed1 th\u1eadp ph\u00e2n th\u1ee9 hai). \u221a B\u00e0i 21. Cho tam gi\u00e1c M N K vu\u00f4ng t\u1ea1i M , \u0111\u01b0\u1eddng cao M I. Bi\u1ebft M N = 4 3cm; N K = 12cm. T\u00ednh N I, M K? B\u00e0i 22. Gi\u1ea3i tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A bi\u1ebft: b = 5cm, C = 30\u25e6. B\u00e0i 23. Gi\u1ea3i tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A c\u00f3 AB = 5cm, B = 40\u25e622\u2032. (K\u1ebft qu\u1ea3 c\u00e1c g\u00f3c l\u00e0m tr\u00f2n \u0111\u1ebfn ph\u00fat, \u0111\u1ed9 d\u00e0i c\u00e1c c\u1ea1nh l\u00e0m tr\u00f2n \u0111\u1ebfn ch\u1eef s\u1ed1 th\u1eadp ph\u00e2n th\u1ee9 hai). B\u00e0i 24. Kh\u00f4ng d\u00f9ng m\u00e1y t\u00ednh b\u1ecf t\u00fai, h\u00e3y s\u1eafp x\u1ebfp c\u00e1c t\u1ec9 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c sau theo th\u1ee9 t\u1ef1 gi\u1ea3m d\u1ea7n: tan 35\u25e6; cot 17\u25e6; tan 83\u25e6; cot 65\u25e6. B\u00e0i 25. S\u1eafp x\u1ebfp c\u00e1c t\u1ec9 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c sau theo th\u1ee9 t\u1ef1 t\u0103ng d\u1ea7n: 1) sin 32\u25e6, cos 32\u25e6, sin 50\u25e6, cos 73\u25e6; | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 63","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m 2) tan 42\u25e6, cot 35\u25e6, tan 67\u25e6, cot 83\u25e6. B\u00e0i 26. Kh\u00f4ng d\u00f9ng m\u00e1y t\u00ednh h\u00e3y s\u1eafp x\u1ebfp c\u00e1c t\u1ec9 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c sau theo th\u1ee9 t\u1ef1 t\u0103ng d\u1ea7n: sin 27\u25e6; cos 31\u25e6; sin 45\u25e6; cos 70\u25e6. B\u00e0i 27. Cho c\u00e1c t\u1ec9 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c sau: sin 60\u25e6; cos 75\u25e6; tan 80\u25e6; cot 52\u25e630\u2032. H\u00e3y vi\u1ebft ch\u00fang th\u00e0nh t\u1ec9 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c c\u1ee7a c\u00e1c g\u00f3c nh\u1ecf h\u01a1n 40\u25e6. B\u00e0i 28. Kh\u00f4ng d\u00f9ng b\u1ea3ng s\u1ed1 v\u00e0 m\u00e1y t\u00ednh b\u1ecf t\u00fai. H\u00e3y s\u1eafp x\u1ebfp c\u00e1c t\u1ec9 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c sau t\u1eeb nh\u1ecf \u0111\u1ebfn l\u1edbn: sin 20\u25e6; cos 20\u25e6; sin 55\u25e6; cos 40\u25e6; tan 70\u25e6. | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT SB\u00e0i 29. Cho \u25b3ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AD v\u00e0 ph\u00e2n gi\u00e1c BE. L\u1ea5y F thu\u1ed9c BC sao cho AF \u22a5 BE t\u1ea1i G. Ch\u1ee9ng minh r\u1eb1ng: 1) BG.BE = BD.BC; 2) \u25b3BGD \u25b3BEC. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 64 N\u0103m h\u1ecdc: 2023 - 2024","Ch\u01b0\u01a1ng 2 \u0110\u01af\u1edcNG TR\u00d2N B\u00e0i 1 \u0110\u01b0\u1eddng tr\u00f2n 1. \u0110\u1ecbnh ngh\u0129a \u0110\u01b0\u1eddng tr\u00f2n t\u00e2m O b\u00e1n k\u00ednh R (R > 0) l\u00e0 t\u1eadp h\u1ee3p g\u1ed3m c\u00e1c \u0111i\u1ec3m c\u00e1ch \u0111i\u1ec3m O | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M m\u1ed9t kho\u1ea3ng b\u1eb1ng R. 2. Ph\u00e2n lo\u1ea1i \u0111\u01b0\u1eddng tr\u00f2n: \u0110\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c: L\u00e0 \u0111\u01b0\u1eddng tr\u00f2n \u0111i qua 3 \u0111\u1ec9nh c\u1ee7a tam gi\u00e1c, t\u00e2m l\u00e0 giao \u0111i\u1ec3m c\u1ee7a 3 \u0111\u01b0\u1eddng trung tr\u1ef1c. \u0110\u01b0\u1eddng tr\u00f2n n\u1ed9i ti\u1ebfp tam gi\u00e1c: L\u00e0 \u0111\u01b0\u1eddng tr\u00f2n ti\u1ebfp x\u00fac v\u1edbi 3 c\u1ea1nh c\u1ee7a tam gi\u00e1c, t\u00e2m l\u00e0 giao \u0111i\u1ec3m c\u1ee7a 3 \u0111\u01b0\u1eddng ph\u00e2n gi\u00e1c trong. \u0110\u01b0\u1eddng tr\u00f2n b\u00e0ng ti\u1ebfp tam gi\u00e1c: L\u00e0 \u0111\u01b0\u1eddng tr\u00f2n ti\u1ebfp x\u00fac v\u1edbi 1 c\u1ea1nh v\u00e0 hai c\u1ea1nh k\u00e9o d\u00e0i c\u1ee7a tam gi\u00e1c, t\u00e2m l\u00e0 giao \u0111i\u1ec3m c\u1ee7a hai \u0111\u01b0\u1eddng ph\u00e2n gi\u00e1c ngo\u00e0i. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 2 M\u1ed9t s\u1ed1 \u0111\u1ecbnh l\u00fd \u0111\u01b0\u1eddng tr\u00f2n I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: 1. \u0110\u1ecbnh l\u00fd tam gi\u00e1c n\u1ed9i ti\u1ebfp: Tam gi\u00e1c n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n l\u00e0 tam gi\u00e1c c\u00f3 3 \u0111\u1ec9nh n\u1eb1m tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n. Tam gi\u00e1c n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n c\u00f3 m\u1ed9t c\u1ea1nh l\u00e0 \u0111\u01b0\u1eddng k\u00ednh th\u00ec tam gi\u00e1c \u0111\u00f3 vu\u00f4ng. Tam gi\u00e1c vu\u00f4ng th\u00ec n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n c\u00f3 t\u00e2m l\u00e0 trung \u0111i\u1ec3m c\u1ea1nh huy\u1ec1n. 2. \u0110\u1ecbnh l\u00fd \u0111\u01b0\u1eddng k\u00ednh - d\u00e2y cung: Trong m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n, n\u1ebfu m\u1ed9t ph\u1ea7n \u0111\u01b0\u1eddng k\u00ednh vu\u00f4ng g\u00f3c v\u1edbi d\u00e2y th\u00ec \u0111i qua trung \u0111i\u1ec3m c\u1ee7a d\u00e2y. | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 65","| C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT | Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m Trong m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n, n\u1ebfu m\u1ed9t ph\u1ea7n \u0111\u01b0\u1eddng k\u00ednh \u0111i qua trung \u0111i\u1ec3m c\u1ee7a d\u00e2y th\u00ec vu\u00f4ng g\u00f3c v\u1edbi d\u00e2y. 3. \u0110\u1ecbnh l\u00fd kho\u1ea3ng c\u00e1ch t\u00e2m - d\u00e2y: Trong m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n, hai d\u00e2y b\u1eb1ng nhau th\u00ec c\u00e1ch \u0111\u1ec1u t\u00e2m v\u00e0 ng\u01b0\u1ee3c l\u1ea1i. Trong m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n, d\u00e2y n\u00e0o l\u1edbn h\u01a1n th\u00ec d\u00e2y \u0111\u00f3 g\u1ea7n t\u00e2m h\u01a1n v\u00e0 ng\u01b0\u1ee3c l\u1ea1i. II. B\u00e0i t\u1eadp: \u0110\u1ecaNH L\u00dd TAM GI\u00c1C N\u1ed8I TI\u1ebeP B\u00e0i 1. Cho (O) c\u00f3 BC l\u00e0 \u0111\u01b0\u1eddng k\u00ednh. Tr\u00ean (O) , l\u1ea5y \u0111i\u1ec3m A. Ch\u1ee9ng minh r\u1eb1ng \u25b3ABC vu\u00f4ng t\u1ea1i A. B\u00e0i 2. Cho (O) c\u00f3 BC l\u00e0 \u0111\u01b0\u1eddng k\u00ednh. Tr\u00ean (O) , l\u1ea5y \u0111i\u1ec3m A v\u00e0 D. Ch\u1ee9ng minh r\u1eb1ng BAC = BDC. B\u00e0i 3. Cho tam gi\u00e1c ABC nh\u1ecdn c\u00f3 hai \u0111\u01b0\u1eddng cao BD v\u00e0 CE. G\u1ecdi O l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BC. 1) Ch\u1ee9ng minh r\u1eb1ng 3 \u0111i\u1ec3m B, E, C c\u00f9ng thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O \u0111\u01b0\u1eddng k\u00ednh BC; 2) Ch\u1ee9ng minh r\u1eb1ng 4 \u0111i\u1ec3m B, E, C, D c\u00f9ng thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O \u0111\u01b0\u1eddng k\u00ednh BC. B\u00e0i 4. Cho (O) c\u00f3 BC l\u00e0 \u0111\u01b0\u1eddng k\u00ednh. Tr\u00ean (O) , l\u1ea5y hai \u0111i\u1ec3m A v\u00e0 D. Ch\u1ee9ng minh r\u1eb1ng c\u00e1c tam gi\u00e1c ABC, DBC l\u00e0 c\u00e1c tam gi\u00e1c vu\u00f4ng. B\u00e0i 5. Cho tam gi\u00e1c \u0111\u1ec1u ABC. G\u1ecdi I v\u00e0 K l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a c\u00e1c c\u1ea1nh AB v\u00e0 AC. 1) Ch\u1ee9ng minh r\u1eb1ng CI v\u00e0 BK l\u00e0 hai \u0111\u01b0\u1eddng cao c\u1ee7a tam gi\u00e1c ABC; 2) G\u1ecdi O l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BC. Ch\u1ee9ng minh r\u1eb1ng 4 \u0111i\u1ec3m B, I, K, C c\u00f9ng thu\u1ed9c m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n. 66 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M B\u00e0i 6. Cho t\u1ee9 gi\u00e1c ABCD c\u00f3 B = D = 90\u25e6. Ch\u1ee9ng minh r\u1eb1ng A, B, C, D c\u00f9ng n\u1eb1m tr\u00ean m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n. B\u00e0i 7. Cho tam gi\u00e1c ABC nh\u1ecdn c\u00f3 hai \u0111\u01b0\u1eddng cao AD, CE. Ch\u1ee9ng minh r\u1eb1ng 4 \u0111i\u1ec3m A, D, C, E c\u00f9ng thu\u1ed9c m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n. B\u00e0i 8. Cho tam gi\u00e1c ABC \u0111\u1ec1u c\u00f3 I v\u00e0 K l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB v\u00e0 AC. Ch\u1ee9ng minh r\u1eb1ng b\u1ed1n \u0111i\u1ec3m B, I, K, C c\u00f9ng thu\u1ed9c m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n. B\u00e0i 9. Cho AC l\u00e0 \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a (O) . V\u1ebd hai d\u00e2y AB v\u00e0 CD song song v\u1edbi nhau. Ch\u1ee9ng minh r\u1eb1ng ba \u0111i\u1ec3m B, O, D th\u1eb3ng h\u00e0ng. \u0110\u1ecaNH L\u00dd M\u1ed8T PH\u1ea6N \u0110\u01af\u1edcNG K\u00cdNH D\u00c2Y CUNG B\u00e0i 1. Cho (O) , \u0111\u01b0\u1eddng k\u00ednh CD. V\u1ebd d\u00e2y AB vu\u00f4ng g\u00f3c v\u1edbi CD t\u1ea1i I. Ch\u1ee9ng minh I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB. B\u00e0i 2. Cho (O) v\u00e0 d\u00e2y AB. G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB. Ch\u1ee9ng minh r\u1eb1ng OI\u22a5AB t\u1ea1i I. B\u00e0i 3. Cho (O; 13cm) v\u00e0 d\u00e2y AB c\u00f3 \u0111\u1ed9 d\u00e0i 10cm. V\u1ebd OI vu\u00f4ng g\u00f3c v\u1edbi AB t\u1ea1i I. T\u00ednh \u0111\u1ed9 d\u00e0i \u0111o\u1ea1n th\u1eb3ng OI. B\u00e0i 4. Cho (O; 6, 5cm) , d\u00e2y AB = 12cm. T\u00ednh kho\u1ea3ng c\u00e1ch t\u1eeb t\u00e2m O \u0111\u1ebfn d\u00e2y AB. B\u00e0i 5. Cho \u0111\u01b0\u1eddng tr\u00f2n (O; 10cm) , d\u00e2y AB = 16cm. V\u1ebd tia OH vu\u00f4ng g\u00f3c v\u1edbi AB t\u1ea1i H. T\u00ednh OH. B\u00e0i 6. Cho (O) , OM l\u00e0 kho\u1ea3ng c\u00e1ch t\u1eeb O \u0111\u1ebfn d\u00e2y AB. Bi\u1ebft OA = 13cm, OM = 5cm. T\u00ednh AB? \u0110\u1ecaNH L\u00dd KHO\u1ea2NG C\u00c1CH T\u1eea T\u00c2M \u0110\u1ebeN D\u00c2Y B\u00e0i 1. Cho \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O c\u00f3 hai d\u00e2y cung AB v\u00e0 CD d\u00e0i b\u1eb1ng nhau v\u00e0 kh\u00f4ng c\u1eaft nhau. K\u1ebb | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 67","| C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT | Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m OH\u22a5AB t\u1ea1i H, OK\u22a5AC t\u1ea1i K. 1) Ch\u1ee9ng minh r\u1eb1ng HA = HB = KC = KD; 2) So s\u00e1nh OH v\u00e0 OK. B\u00e0i 2. Cho \u0111\u01b0\u1eddng tr\u00f2n (O) , hai d\u00e2y AB v\u00e0 CD. Tr\u00ean AB l\u1ea5y I, tr\u00ean CD l\u1ea5y K sao cho: OI\u22a5AB t\u1ea1i I; OK\u22a5CD t\u1ea1i K; OI > OK. H\u00e3y so s\u00e1nh 2 d\u00e2y AB v\u00e0 CD. B\u00e0i 3. Cho \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O c\u00f3 hai d\u00e2y AB v\u00e0 CD b\u1eb1ng nhau c\u1eaft nhau t\u1ea1i I. G\u1ecdi H v\u00e0 K l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB v\u00e0 CD. 1) So s\u00e1nh OH v\u00e0 OK; 2) Ch\u1ee9ng minh r\u1eb1ng IH = IK. B\u00e0i 4. Cho h\u00ecnh b\u00ean d\u01b0\u1edbi, trong \u0111\u00f3 c\u00f3 hai \u0111\u01b0\u1eddng tr\u00f2n c\u00f3 c\u00f9ng t\u00e2m l\u00e0 O. Bi\u1ebft EF > IJ. So s\u00e1nh AD v\u00e0 HD. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 3 Luy\u1ec7n t\u1eadp: \u0110\u01b0\u1eddng tr\u00f2n v\u00e0 m\u1ed9t s\u1ed1 \u0111\u1ecbnh l\u00fd B\u00e0i 1. Cho (O) c\u00f3 hai d\u00e2y AB, CD kh\u00f4ng c\u1eaft nhau. G\u1ecdi H l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB. T\u1eeb O k\u1ebb OK\u22a5CD t\u1ea1i K. 1) Ch\u1ee9ng minh r\u1eb1ng OH\u22a5AB; 2) Ch\u1ee9ng minh r\u1eb1ng K l\u00e0 trung \u0111i\u1ec3m c\u1ee7a CD. B\u00e0i 2. Cho \u0111\u01b0\u1eddng tr\u00f2n (O; 10cm) , d\u00e2y AB = 16cm. V\u1ebd tia OH vu\u00f4ng g\u00f3c v\u1edbi AB t\u1ea1i H. T\u00ednh OH. B\u00e0i 3. Cho (O; 13cm) v\u00e0 kho\u1ea3ng c\u00e1ch t\u1eeb t\u00e2m O \u0111\u1ebfn d\u00e2y M N l\u00e0 9cm. T\u00ednh \u0111\u1ed9 d\u00e0i d\u00e2y M N. B\u00e0i 4. Cho (O; 5cm) v\u00e0 d\u00e2y AB = 8cm. K\u1ebb tia Ox\u22a5AB t\u1ea1i I v\u00e0 c\u1eaft \u0111\u01b0\u1eddng tr\u00f2n t\u1ea1i M. T\u00ednh M I. \u221a B\u00e0i 5. Cho AB l\u00e0 m\u1ed9t d\u00e2y c\u1ee7a (O; 2cm) , bi\u1ebft AB = 2 3cm. T\u00ednh s\u1ed1 \u0111o AOB ? 68 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M B\u00e0i 6. Cho (O) \u0111\u01b0\u1eddng k\u00ednh AB v\u00e0 d\u00e2y CD kh\u00f4ng c\u1eaft AB (C n\u1eb1m gi\u1eefa A v\u00e0 D tr\u00ean (O)). V\u1ebd OI, AH, BK c\u00f9ng vu\u00f4ng g\u00f3c v\u1edbi CD l\u1ea7n l\u01b0\u1ee3t \u1edf I, H v\u00e0 K. 1) Ch\u1ee9ng minh r\u1eb1ng I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a HK; 2) Ch\u1ee9ng minh r\u1eb1ng CH = DK. B\u00e0i 7. Cho (O; R) , H l\u00e0 \u0111i\u1ec3m b\u00ean trong \u0111\u01b0\u1eddng tr\u00f2n (H kh\u00f4ng tr\u00f9ng v\u1edbi O). V\u1ebd \u0111\u01b0\u1eddng k\u00ednh AB qua H (HB < HA) . V\u1ebd d\u00e2y CD vu\u00f4ng g\u00f3c v\u1edbi AB t\u1ea1i H. Ch\u1ee9ng minh r\u1eb1ng: 1) BCA = 90\u25e6; 2) CH.HD = HB.HA. B\u00e0i 8. Cho tam gi\u00e1c ABC nh\u1ecdn c\u00f3 hai \u0111\u01b0\u1eddng cao BD v\u00e0 CE. G\u1ecdi O l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BC. 1) Ch\u1ee9ng minh r\u1eb1ng 4 \u0111i\u1ec3m B, E, C, D c\u00f9ng thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O \u0111\u01b0\u1eddng k\u00ednh BC; 2) G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a DE. Ch\u1ee9ng minh r\u1eb1ng OI\u22a5ED. B\u00e0i 9. Cho tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp (O) sao cho t\u00e2m O n\u1eb1m trong tam gi\u00e1c. G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB, k\u1ebb M K\u22a5BC. Bi\u1ebft AB = 30cm, M K = 12cm, BC = 36cm. T\u00ednh AC v\u00e0 b\u00e1n k\u00ednh c\u1ee7a (O) . B\u00e0i 10. Cho (O) \u0111\u01b0\u1eddng k\u00ednh AB, d\u00e2y cung CD kh\u00f4ng c\u1eaft \u0111\u01b0\u1eddng k\u00ednh AB. G\u1ecdi H, K theo th\u1ee9 t\u1ef1 l\u00e0 ch\u00e2n c\u00e1c \u0111\u01b0\u1eddng vu\u00f4ng g\u00f3c k\u1ebb t\u1eeb A v\u00e0 B \u0111\u1ebfn CD. Ch\u1ee9ng minh r\u1eb1ng CH = DK. B\u00e0i 11. Cho \u0111i\u1ec3m A thu\u1ed9c (O) \u0111\u01b0\u1eddng k\u00ednh BC. V\u1ebd AH\u22a5BC \u1edf H n\u1eb1m gi\u1eefa O v\u00e0 B. V\u1ebd \u0111\u01b0\u1eddng k\u00ednh AD. 1) Ch\u1ee9ng minh r\u1eb1ng AB.AC = AD.AH; 2) Ch\u1ee9ng minh r\u1eb1ng CAH = BAD. B\u00e0i 12. Cho tam gi\u00e1c ABC nh\u1ecdn n\u1ed9i ti\u1ebfp (O) v\u00e0 c\u00f3 tr\u1ef1c t\u00e2m H. V\u1ebd \u0111\u01b0\u1eddng k\u00ednh AK. 1) Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c BHCK l\u00e0 h\u00ecnh b\u00ecnh h\u00e0nh; 2) K\u1ebb OM \u22a5BC \u1edf M. Ch\u1ee9ng minh r\u1eb1ng ba \u0111i\u1ec3m H, M, K th\u1eb3ng h\u00e0ng; 3) Ch\u1ee9ng minh r\u1eb1ng AH = 2OM. B\u00e0i 13. Cho (O) , \u0111\u01b0\u1eddng k\u00ednh AB = 2R. G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AO, qua I k\u1ebb d\u00e2y CD vu\u00f4ng g\u00f3c v\u1edbi OA. 1) T\u1ee9 gi\u00e1c ACOD l\u00e0 h\u00ecnh g\u00ec? V\u00ec sao?; 2) Ch\u1ee9ng minh r\u1eb1ng tam gi\u00e1c BCD \u0111\u1ec1u. B\u00e0i 14. Cho tam gi\u00e1c ABC nh\u1ecdn c\u00f3 hai \u0111\u01b0\u1eddng cao BD v\u00e0 CE. G\u1ecdi O v\u00e0 I l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BC v\u00e0 DE. Ch\u1ee9ng minh r\u1eb1ng OI vu\u00f4ng g\u00f3c v\u1edbi DE. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 69","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m B\u00e0i 4 Ti\u1ebfp tuy\u1ebfn v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: 1. \u0110\u1ecbnh ngh\u0129a: Ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng ch\u1ec9 c\u00f3 m\u1ed9t \u0111i\u1ec3m chung v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n. \u0110i\u1ec3m chung \u0111\u00f3 g\u1ecdi l\u00e0 ti\u1ebfp \u0111i\u1ec3m. 2. \u0110\u1ecbnh l\u00fd: N\u1ebfu m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n th\u00ec n\u00f3 vu\u00f4ng g\u00f3c v\u1edbi b\u00e1n k\u00ednh \u0111i qua ti\u1ebfp \u0111i\u1ec3m. N\u1ebfu m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi b\u00e1n k\u00ednh t\u1ea1i m\u1ed9t \u0111i\u1ec3m thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n th\u00ec \u0111\u01b0\u1eddng th\u1eb3ng \u0111\u00f3 l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n. II. B\u00e0i t\u1eadp: B\u00e0i 1. T\u1eeb \u0111i\u1ec3m M n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n (O; 9cm) , k\u1ebb ti\u1ebfp tuy\u1ebfn M A v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (A l\u00e0 ti\u1ebfp \u0111i\u1ec3m). Bi\u1ebft M OA = 30\u25e6. T\u00ednh \u0111\u1ed9 d\u00e0i \u0111o\u1ea1n th\u1eb3ng AM. B\u00e0i 2. Cho \u0111\u01b0\u1eddng tr\u00f2n (O; 3cm) , t\u1eeb \u0111i\u1ec3m A n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n k\u1ebb ti\u1ebfp tuy\u1ebfn AB (B l\u00e0 ti\u1ebfp \u0111i\u1ec3m). Bi\u1ebft AB = 4cm. T\u00ednh kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn O. B\u00e0i 3. Cho \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O c\u00f3 b\u00e1n k\u00ednh b\u1eb1ng 5cm v\u00e0 \u0111i\u1ec3m B c\u00e1ch O m\u1ed9t kho\u1ea3ng 13cm. L\u1ea5y A thu\u1ed9c (O) sao cho AB = 12cm. 1) Tam gi\u00e1c OAB l\u00e0 tam gi\u00e1c g\u00ec?; 2) Ch\u1ee9ng minh r\u1eb1ng \u0111\u01b0\u1eddng th\u1eb3ng AB ti\u1ebfp x\u00fac v\u1edbi (O) . B\u00e0i 4. T\u1eeb \u0111i\u1ec3m A n\u1eb1m ngo\u00e0i (O) , v\u1ebd ti\u1ebfp tuy\u1ebfn v\u1edbi ti\u1ebfp \u0111i\u1ec3m B. L\u1ea5y C thu\u1ed9c (O) sao cho AC = AB. Ch\u1ee9ng minh r\u1eb1ng AC l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a (O) . B\u00e0i 5. Cho \u0111i\u1ec3m A b\u00ean ngo\u00e0i (O) , \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m I \u0111\u01b0\u1eddng k\u00ednh AO c\u1eaft (O) t\u1ea1i hai \u0111i\u1ec3m B v\u00e0 C. Ch\u1ee9ng minh r\u1eb1ng AB v\u00e0 AC l\u00e0 hai ti\u1ebfp tuy\u1ebfn c\u1ee7a (O) . B\u00e0i 6. T\u1eeb m\u1ed9t \u0111i\u1ec3m B n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O b\u00e1n k\u00ednh 9cm, k\u1ebb ti\u1ebfp tuy\u1ebfn BA v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (A l\u00e0 ti\u1ebfp \u0111i\u1ec3m). K\u1ebb \u0111\u01b0\u1eddng cao OH c\u1ee7a tam gi\u00e1c OAB (H thu\u1ed9c OB). Bi\u1ebft OH = 5, 4cm. T\u00ednh OA, OB. B\u00e0i 7. Cho \u25b3ABC vu\u00f4ng \u1edf A c\u00f3 ABC = 30\u25e6, AB = 4cm. Ch\u1ee9ng minh r\u1eb1ng BC ti\u1ebfp x\u00fac v\u1edbi (A; 2cm) . B\u00e0i 8. Cho (O) v\u00e0 d\u00e2y AB kh\u00e1c \u0111\u01b0\u1eddng k\u00ednh. V\u1ebd tia Ax sao cho AB l\u00e0 ph\u00e2n gi\u00e1c c\u1ee7a OAx. Qua B v\u1ebd BM vu\u00f4ng g\u00f3c v\u1edbi tia Ax t\u1ea1i M. Ch\u1ee9ng minh r\u1eb1ng \u0111\u01b0\u1eddng th\u1eb3ng BM l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a 70 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 (O) . B\u00e0i 9. Cho tam gi\u00e1c ABC v\u1ebd \u0111\u01b0\u1eddng cao AH (H n\u1eb1m gi\u1eefa B v\u00e0 C). Bi\u1ebft AH2 = HB.HC. Ch\u1ee9ng minh r\u1eb1ng \u0111\u01b0\u1eddng th\u1eb3ng AC l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m B, b\u00e1n k\u00ednh BA. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 5 Hai ti\u1ebfp tuy\u1ebfn c\u1eaft nhau I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M N\u1ebfu hai ti\u1ebfp tuy\u1ebfn c\u1ee7a m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n c\u1eaft nhau th\u00ec: Giao \u0111i\u1ec3m c\u00e1ch \u0111\u1ec1u hai ti\u1ebfp \u0111i\u1ec3m. \u0110\u01b0\u1eddng th\u1eb3ng n\u1ed1i t\u00e2m v\u00e0 giao \u0111i\u1ec3m l\u00e0 tia ph\u00e2n gi\u00e1c c\u1ee7a g\u00f3c t\u1ea1o b\u1edfi hai ti\u1ebfp tuy\u1ebfn v\u00e0 l\u00e0 tia ph\u00e2n gi\u00e1c c\u1ee7a g\u00f3c t\u1ea1o b\u1edfi hai b\u00e1n k\u00ednh \u0111i qua hai ti\u1ebfp \u0111i\u1ec3m. II. B\u00e0i t\u1eadp: B\u00e0i 1. Cho (O; 6cm) v\u00e0 \u0111i\u1ec3m A c\u00e1ch O m\u1ed9t kho\u1ea3ng b\u1eb1ng 12cm. V\u1ebd ti\u1ebfp tuy\u1ebfn AB, AC v\u1edbi (O) (B, C l\u00e0 ti\u1ebfp \u0111i\u1ec3m). T\u00ednh s\u1ed1 \u0111o BOC. B\u00e0i 2. Cho (O; 5cm) v\u00e0 \u0111i\u1ec3m A c\u00e1ch O m\u1ed9t kho\u1ea3ng b\u1eb1ng 10cm. V\u1ebd ti\u1ebfp tuy\u1ebfn AB, AC v\u1edbi (O) (B, C l\u00e0 ti\u1ebfp \u0111i\u1ec3m). T\u00ednh s\u1ed1 \u0111o BAC. B\u00e0i 3. Cho AC l\u00e0 \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a (O) . Tr\u00ean ti\u1ebfp tuy\u1ebfn t\u1ea1i A c\u1ee7a (O) l\u1ea5y \u0111i\u1ec3m I, v\u1ebd d\u00e2y cung | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 71","| C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT | Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m CB song song v\u1edbi OI. Ch\u1ee9ng minh r\u1eb1ng: 1) IOA = IOB; 2) IB l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a (O) . B\u00e0i 4. Hai ti\u1ebfp tuy\u1ebfn t\u1ea1i B v\u00e0 C c\u1ee7a (O) c\u1eaft nhau \u1edf A. T\u1eeb O k\u1ebb tia vu\u00f4ng g\u00f3c v\u1edbi OB c\u1eaft AC t\u1ea1i D. 1) Ch\u1ee9ng minh r\u1eb1ng OD\/\/AB; 2) Ch\u1ee9ng minh r\u1eb1ng DO = DA. B\u00e0i 5. Cho (O) c\u00f3 \u0111i\u1ec3m A n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n. K\u1ebb c\u00e1c ti\u1ebfp tuy\u1ebfn AB v\u00e0 AC v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (B, C l\u00e0 c\u00e1c ti\u1ebfp \u0111i\u1ec3m). 1) Ch\u1ee9ng minh r\u1eb1ng BC vu\u00f4ng g\u00f3c v\u1edbi OA; 2) K\u1ebb \u0111\u01b0\u1eddng k\u00ednh BD, Ch\u1ee9ng minh r\u1eb1ng OA\/\/CD. B\u00e0i 6. T\u1eeb m\u1ed9t \u0111i\u1ec3m A n\u1eb1m b\u00ean ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n (O) , k\u1ebb c\u00e1c ti\u1ebfp tuy\u1ebfn AB, AC v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (B, C l\u00e0 c\u00e1c ti\u1ebfp \u0111i\u1ec3m). Qua \u0111i\u1ec3m M thu\u1ed9c cung nh\u1ecf BC, k\u1ebb ti\u1ebfp tuy\u1ebfn v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (O) , n\u00f3 c\u1eaft c\u00e1c ti\u1ebfp tuy\u1ebfn AB v\u00e0 AC theo th\u1ee9 t\u1ef1 \u1edf D v\u00e0 E. Ch\u1ee9ng minh r\u1eb1ng chu vi tam gi\u00e1c ADE b\u1eb1ng 2AB. B\u00e0i 7. Cho n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O \u0111\u01b0\u1eddng k\u00ednh AB. G\u1ecdi Ax, By l\u00e0 c\u00e1c tia vu\u00f4ng g\u00f3c v\u1edbi AB (Ax, By v\u00e0 n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n c\u00f9ng thu\u1ed9c m\u1ed9t n\u1eeda m\u1eb7t ph\u1eb3ng b\u1edd AB). Qua \u0111i\u1ec3m M thu\u1ed9c n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n (M kh\u00e1c A v\u00e0 B), k\u1ebb ti\u1ebfp tuy\u1ebfn v\u1edbi n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n, n\u00f3 c\u1eaft Ax v\u00e0 By theo th\u1ee9 t\u1ef1 \u1edf C v\u00e0 D. Ch\u1ee9ng minh r\u1eb1ng: 1) COD = 90\u25e6; 2) CD = AC + BD; 3) T\u00edch AC.BD kh\u00f4ng \u0111\u1ed5i khi \u0111i\u1ec3m M di chuy\u1ec3n tr\u00ean n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n. B\u00e0i 8. Cho h\u00ecnh v\u1ebd b\u00ean, bi\u1ebft tam gi\u00e1c ABC ngo\u1ea1i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n (O) . 1) Ch\u1ee9ng minh r\u1eb1ng: 2AD = AB + AC\u02d8BC; 2) T\u00ecm c\u00e1c h\u1ec7 th\u1ee9c t\u01b0\u01a1ng t\u1ef1 nh\u01b0 h\u1ec7 th\u1ee9c \u1edf c\u00e2u 1). \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 72 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 B\u00e0i 6 Luy\u1ec7n t\u1eadp: Ti\u1ebfp tuy\u1ebfn v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n B\u00e0i 1. T\u1eeb \u0111i\u1ec3m M n\u1eb1m ngo\u00e0i (O) v\u1ebd hai ti\u1ebfp tuy\u1ebfn M A v\u00e0 M B sao cho AM B = 60\u25e6. Bi\u1ebft chu | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M vi tam gi\u00e1c M AB l\u00e0 8cm, t\u00ednh \u0111\u1ed9 d\u00e0i d\u00e2y cung AB. B\u00e0i 2. Cho tam gi\u00e1c ABC v\u1ebd \u0111\u01b0\u1eddng cao AH (H n\u1eb1m gi\u1eefa B v\u00e0 C). Bi\u1ebft AH2 = HB.HC. Ch\u1ee9ng minh r\u1eb1ng \u0111\u01b0\u1eddng th\u1eb3ng AC l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m B, b\u00e1n k\u00ednh BA. B\u00e0i 3. Cho (O; 15cm) , d\u00e2y BC = 24cm. C\u00e1c ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n t\u1ea1i B v\u00e0 C c\u1eaft nhau \u1edf A. T\u00ednh \u0111\u1ed9 d\u00e0i OA, AB, AC. B\u00e0i 4. Cho \u25b3ABC c\u00e2n t\u1ea1i B (AB < BC) . (O) \u0111\u01b0\u1eddng k\u00ednh AC c\u1eaft BC t\u1ea1i H. Ch\u1ee9ng minh r\u1eb1ng AH \u22a5B C. B\u00e0i 5. Cho (O) . T\u1eeb \u0111i\u1ec3m A n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n k\u1ebb c\u00e1c ti\u1ebfp tuy\u1ebfn AB, AC v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (B, C l\u00e0 ti\u1ebfp \u0111i\u1ec3m). Bi\u1ebft OB = 2cm, OA = 4cm. T\u00ednh chu vi tam gi\u00e1c ABC. B\u00e0i 6. Cho h\u00ecnh thang vu\u00f4ng ABCD vu\u00f4ng t\u1ea1i A v\u00e0 D. Bi\u1ebft BM C = 90\u25e6 v\u1edbi M l\u00e0 trung \u0111i\u1ec3m AD. Ch\u1ee9ng minh r\u1eb1ng AD l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1eddng k\u00ednh BC. B\u00e0i 7. Hai ti\u1ebfp tuy\u1ebfn t\u1ea1i A v\u00e0 B c\u1ee7a (O) c\u1eaft nhau \u1edf I. \u0110\u01b0\u1eddng th\u1eb3ng qua I vu\u00f4ng g\u00f3c v\u1edbi IA c\u1eaft tia OB t\u1ea1i K. Ch\u1ee9ng minh r\u1eb1ng: 1) IK\/\/OA; 2) Tam gi\u00e1c IOK c\u00e2n. B\u00e0i 8. Cho \u0111\u01b0\u1eddng th\u1eb3ng d kh\u00f4ng c\u1eaft (O) , v\u1ebd OA vu\u00f4ng g\u00f3c v\u1edbi d t\u1ea1i A. T\u1eeb m\u1ed9t \u0111i\u1ec3m M thu\u1ed9c \u0111\u01b0\u1eddng th\u1eb3ng d, (M =\u0338 A) . V\u1ebd hai ti\u1ebfp tuy\u1ebfn M B, M C v\u1edbi (O) . Ch\u1ee9ng minh r\u1eb1ng 5 \u0111i\u1ec3m A, B, C, O, M c\u00f9ng thu\u1ed9c m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n. B\u00e0i 9. T\u1eeb m\u1ed9t \u0111i\u1ec3m M n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O, v\u1ebd hai ti\u1ebfp tuy\u1ebfn M A v\u00e0 M B v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (A, B l\u00e0 ti\u1ebfp \u0111i\u1ec3m). G\u1ecdi I l\u00e0 giao \u0111i\u1ec3m c\u1ee7a OM v\u00e0 (O) . Ti\u1ebfp tuy\u1ebfn t\u1ea1i I v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n 1 l\u1ea7n l\u01b0\u1ee3t c\u1eaft M A, M B t\u1ea1i E v\u00e0 F. Ch\u1ee9ng minh r\u1eb1ng EOF = AOB. 2 B\u00e0i 10. Cho n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O, \u0111\u01b0\u1eddng k\u00ednh AB. K\u1ebb ti\u1ebfp tuy\u1ebfn Ax, By c\u00f9ng ph\u00eda v\u1edbi n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n \u0111\u1ed1i v\u1edbi AB. V\u1ebd b\u00e1n k\u00ednh OE b\u1ea5t k\u1ef3. Ti\u1ebfp tuy\u1ebfn n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n t\u1ea1i E c\u1eaft Ax v\u00e0 By l\u1ea7n l\u01b0\u1ee3t t\u1ea1i C v\u00e0 D. 1) Ch\u1ee9ng minh r\u1eb1ng CD = AC + BD; 2) T\u00ednh s\u1ed1 \u0111o DOC. B\u00e0i 11. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A, \u0111\u01b0\u1eddng cao AH. V\u1ebd \u0111\u01b0\u1eddng tr\u00f2n (A; AH) . K\u1ebb c\u00e1c ti\u1ebfp | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 73","| C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT | Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m tuy\u1ebfn BD, CE, v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (D, E l\u00e0 c\u00e1c ti\u1ebfp \u0111i\u1ec3m kh\u00e1c H). Ch\u1ee9ng minh r\u1eb1ng: 1) BD + CE = BC; 2) Ba \u0111i\u1ec3m D, A, E th\u1eb3ng h\u00e0ng; 3) DE l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1eddng k\u00ednh BC. B\u00e0i 12. T\u1eeb \u0111i\u1ec3m A n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n (O) , k\u1ebb hai ti\u1ebfp tuy\u1ebfn AB, AC (B, C l\u00e0 hai ti\u1ebfp \u0111i\u1ec3m) sao cho BAC \u0338= 90\u25e6. K\u1ebb tia Ax vu\u00f4ng g\u00f3c v\u1edbi AB v\u00e0 c\u1eaft \u0111\u01b0\u1eddng th\u1eb3ng OC t\u1ea1i D. Ch\u1ee9ng minh r\u1eb1ng DA = DO. B\u00e0i 13. T\u1eeb m\u1ed9t \u0111i\u1ec3m B n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O b\u00e1n k\u00ednh 9cm, k\u1ebb ti\u1ebfp tuy\u1ebfn BA v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (A l\u00e0 ti\u1ebfp \u0111i\u1ec3m). K\u1ebb \u0111\u01b0\u1eddng cao AH c\u1ee7a \u25b3OAB (H thu\u1ed9c OB). Bi\u1ebft OH = 5, 4cm. T\u00ednh AB, OB. B\u00e0i 14. Cho (O; 6cm) v\u00e0 \u0111i\u1ec3m A c\u00e1ch O m\u1ed9t kho\u1ea3ng b\u1eb1ng 12cm. V\u1ebd ti\u1ebfp tuy\u1ebfn AB, AC v\u1edbi (O) , (B, C l\u00e0 ti\u1ebfp \u0111i\u1ec3m. T\u00ednh s\u1ed1 \u0111o BOC. B\u00e0i 15. Cho (O; 5cm) v\u00e0 d\u00e2y AB c\u00e1ch O m\u1ed9t kho\u1ea3ng 10cm. K\u1ebb c\u00e1c ti\u1ebfp tuy\u1ebfn AB, AC v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (B, C l\u00e0 ti\u1ebfp \u0111i\u1ec3m). T\u00ednh BAC. B\u00e0i 16. Cho tam gi\u00e1c ABC c\u00e2n t\u1ea1i A, AB = 12cm, \u0111\u01b0\u1eddng cao AH = 8cm. T\u00ednh b\u00e1n k\u00ednh c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c ABC. B\u00e0i 17. Cho (O; 5cm) v\u00e0 \u0111i\u1ec3m A c\u00e1ch O m\u1ed9t kho\u1ea3ng b\u1eb1ng 10cm. K\u1ebb c\u00e1c ti\u1ebfp tuy\u1ebfn AB, AC v\u1edbi (O) . T\u00ednh BAC. B\u00e0i 18. Tam gi\u00e1c c\u00f3 \u0111\u1ed9 d\u00e0i ba c\u1ea1nh l\u00e0 5cm, 12cm, 13cm. T\u00ednh b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c. B\u00e0i 19. Cho (O) \u0111\u01b0\u1eddng k\u00ednh AB. \u0110\u01b0\u1eddng th\u1eb3ng xy ti\u1ebfp x\u00fac v\u1edbi (O) t\u1ea1i A, l\u1ea5y K thu\u1ed9c (O) v\u00e0 kh\u00e1c A, B. V\u1ebd KH vu\u00f4ng g\u00f3c v\u1edbi AB t\u1ea1i H v\u00e0 I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a KH. Tia BI c\u1eaft xy t\u1ea1i M, BK c\u1eaft xy t\u1ea1i C. 1) Ch\u1ee9ng minh r\u1eb1ng M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AC; 2) Ch\u1ee9ng minh r\u1eb1ng OM\/\/BK, AK\u22a5OM ; 3) Ch\u1ee9ng minh r\u1eb1ng M K ti\u1ebfp x\u00fac v\u1edbi (O) t\u1ea1i K. B\u00e0i 20. Cho n\u1eeda (O) \u0111\u01b0\u1eddng k\u00ednh AB. V\u1ebd hai ti\u1ebfp tuy\u1ebfn Ax v\u00e0 By \u1edf c\u00f9ng n\u1eeda m\u1eb7t ph\u1eb3ng ch\u1ee9a n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n. Ti\u1ebfp tuy\u1ebfn t\u1ea1i M c\u1ee7a n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n c\u1eaft Ax v\u00e0 By l\u1ea7n l\u01b0\u1ee3t \u1edf C v\u00e0 D. Ch\u1ee9ng minh r\u1eb1ng: 1) AC + BD = CD; 2) \u25b3COD vu\u00f4ng \u1edf O; 3) AC.BD = R2 B\u00e0i 21. T\u1eeb \u0111i\u1ec3m M n\u1eb1m ngo\u00e0i (O) , v\u1ebd hai ti\u1ebfp tuy\u1ebfn M A, M B v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (A, B l\u00e0 ti\u1ebfp 74 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 \u0111i\u1ec3m). G\u1ecdi H l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AB v\u00e0 OM. V\u1ebd \u0111\u01b0\u1eddng k\u00ednh BD c\u1ee7a (O) . Ch\u1ee9ng minh r\u1eb1ng BD2 = 4OH.OM. B\u00e0i 22. Cho \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O \u0111\u01b0\u1eddng k\u00ednh AB v\u00e0 m\u1ed9t \u0111i\u1ec3m C tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n. T\u1eeb O k\u1ebb m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng song song v\u1edbi d\u00e2y AC, \u0111\u01b0\u1eddng th\u1eb3ng n\u00e0y c\u1eaft ti\u1ebfp tuy\u1ebfn t\u1ea1i B c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n \u1edf \u0111i\u1ec3m D. 1) Ch\u1ee9ng minh r\u1eb1ng OD l\u00e0 ph\u00e2n gi\u00e1c c\u1ee7a BOC; 2) Ch\u1ee9ng minh r\u1eb1ng CD l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n. B\u00e0i 23. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A (AB < AC). \u0110\u01b0\u1eddng tr\u00f2n (O) \u0111\u01b0\u1eddng k\u00ednh AC c\u1eaft BC | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M t\u1ea1i H. 1) Ch\u1ee9ng minh r\u1eb1ng AH\u22a5BC; 2) G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB. Ch\u1ee9ng minh r\u1eb1ng HM l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a (O) . B\u00e0i 24. Cho n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n (O) \u0111\u01b0\u1eddng k\u00ednh AB. C l\u00e0 m\u1ed9t \u0111i\u1ec3m tr\u00ean n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n, \u0111\u01b0\u1eddng th\u1eb3ng d ti\u1ebfp x\u00fac v\u1edbi n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n t\u1ea1i C. T\u1eeb A v\u00e0 B k\u1ebb AD v\u00e0 BE vu\u00f4ng g\u00f3c v\u1edbi d. K\u1ebb CH vu\u00f4ng g\u00f3c v\u1edbi AB. Ch\u1ee9ng minh r\u1eb1ng CH2 = AD.BE. B\u00e0i 25. Cho (O; 3cm) . T\u1eeb m\u1ed9t \u0111i\u1ec3m A n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n, k\u1ebb ti\u1ebfp t\u221auy\u1ebfn AB v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n 33 (B l\u00e0 ti\u1ebfp \u0111i\u1ec3m). K\u1ebb \u0111\u01b0\u1eddng cao BH c\u1ee7a tam gi\u00e1c AOB, bi\u1ebft BH = cm. T\u00ednh OA. 2 B\u00e0i 26. Cho n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n (O; R) \u0111\u01b0\u1eddng k\u00ednh AB, b\u00e1n k\u00ednh OC vu\u00f4ng g\u00f3c AB. G\u1ecdi M l\u00e0 \u0111i\u1ec3m tr\u00ean n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n (M kh\u00f4ng tr\u00f9ng A v\u00e0 B). Ti\u1ebfp tuy\u1ebfn t\u1ea1i M c\u1eaft ti\u1ebfp tuy\u1ebfn t\u1ea1i A c\u1ee7a n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00f3 t\u1ea1i E v\u00e0 c\u1eaft OC t\u1ea1i D. Bi\u1ebft r\u1eb1ng AE c\u1eaft BD t\u1ea1i F. Ch\u1ee9ng minh r\u1eb1ng EA.EF = R2. B\u00e0i 27. Cho (O; 5cm) , \u0111\u01b0\u1eddng k\u00ednh AB. G\u1ecdi E l\u00e0 m\u1ed9t \u0111i\u1ec3m tr\u00ean AB sao cho BE = 2cm. Qua trung \u0111i\u1ec3m H c\u1ee7a \u0111o\u1ea1n AE v\u1ebd d\u00e2y cung CD vu\u00f4ng g\u00f3c v\u1edbi AB. 1) T\u1ee9 gi\u00e1c ACED l\u00e0 h\u00ecnh g\u00ec? V\u00ec sao?; 2) G\u1ecdi I l\u00e0 giao \u0111i\u1ec3m c\u1ee7a DE v\u1edbi BC. Ch\u1ee9ng minh r\u1eb1ng I thu\u1ed9c (O) \u0111\u01b0\u1eddng k\u00ednh EB; 3) Ch\u1ee9ng minh HI l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a (O); 4) T\u00ednh HI. B\u00e0i 28. G\u1ecdi C l\u00e0 m\u1ed9t \u0111i\u1ec3m b\u1ea5t k\u1ef3 n\u1eb1m tr\u00ean n\u1eeda (O) \u0111\u01b0\u1eddng k\u00ednh AB b\u1eb1ng 2R C \u0338= A, C =\u0338 B. Tia BC c\u1eaft ti\u1ebfp tuy\u1ebfn t\u1ea1i A c\u1ee7a n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n \u1edf M. Ti\u1ebfp tuy\u1ebfn t\u1ea1i C c\u1ee7a n\u1eeda (O) c\u1eaft AM t\u1ea1i I. 1) Ch\u1ee9ng minh r\u1eb1ng 4 \u0111i\u1ec3m I; A; O; C c\u00f9ng n\u1eb1m tr\u00ean m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n; 2) Ch\u1ee9ng minh r\u1eb1ng IC2 = 1 M B.M C. 4 B\u00e0i 29. Cho tam gi\u00e1c M AB, v\u1ebd \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O \u0111\u01b0\u1eddng k\u00ednh AB c\u1eaft M A \u1edf C c\u1eaft M B \u1edf D. K\u1ebb AP \u22a5CD, BQ\u22a5CD. G\u1ecdi H l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AD v\u00e0 BC. Ch\u1ee9ng minh r\u1eb1ng: | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 75","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m 1) P D.DQ = P A.BQ; 2) QC.CP = P D.QD; 3) M H\u22a5AB. B\u00e0i 30. Cho \u0111\u01b0\u1eddng tr\u00f2n (O) , \u0111i\u1ec3m A n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n. K\u1ebb c\u00e1c ti\u1ebfp tuy\u1ebfn AB, AC v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (B, C l\u00e0 c\u00e1c ti\u1ebfp \u0111i\u1ec3m). K\u1ebb \u0111\u01b0\u1eddng k\u00ednh BD, ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n (O) t\u1ea1i D c\u1eaft \u0111\u01b0\u1eddng th\u1eb3ng BC t\u1ea1i E. Ch\u1ee9ng minh r\u1eb1ng \u25b3OCE \u25b3ACD. B\u00e0i 31. Cho (O) , \u0111\u01b0\u1eddng k\u00ednh AB, d\u00e2y CD vu\u00f4ng g\u00f3c v\u1edbi OA t\u1ea1i \u0111i\u1ec3m H n\u1eb1m gi\u1eefa O v\u00e0 A. G\u1ecdi E l\u00e0 \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9ng v\u1edbi A qua H. Ch\u1ee9ng minh r\u1eb1ng CH2 + CE2 \u2212 3HA2 = 2.HE.EB. B\u00e0i 32. Cho tam gi\u00e1c ABC ngo\u1ea1i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n (O) . G\u1ecdi I l\u00e0 ti\u1ebfp \u0111i\u1ec3m c\u1ee7a BC v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (O) . Bi\u1ebft AB.AC = 2IB.IC. T\u00ednh s\u1ed1 \u0111o A. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT SB\u00e0i 7 V\u1ecb tr\u00ed t\u01b0\u01a1ng \u0111\u1ed1i c\u1ee7a hai \u0111\u01b0\u1eddng tr\u00f2n I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: Cho hai \u0111\u01b0\u1eddng tr\u00f2n (O; R) v\u00e0 (O\u2032; R\u2032): N\u1ebfu hai \u0111\u01b0\u1eddng tr\u00f2n n\u1eb1m ngo\u00e0i nhau th\u00ec: OO\u2032 > R + R\u2032. N\u1ebfu hai \u0111\u01b0\u1eddng tr\u00f2n ti\u1ebfp x\u00fac ngo\u00e0i th\u00ec: OO\u2032 = R + R\u2032 (Ti\u1ebfp \u0111i\u1ec3m n\u1eb1m tr\u00ean \u0111o\u1ea1n n\u1ed1i t\u00e2m). N\u1ebfu hai \u0111\u01b0\u1eddng tr\u00f2n c\u1eaft nhau th\u00ec: R \u2212 R\u2032 < OO\u2032 < R + R\u2032 (\u0110\u01b0\u1eddng th\u1eb3ng n\u1ed1i t\u00e2m l\u00e0 \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a d\u00e2y cung chung). N\u1ebfu hai \u0111\u01b0\u1eddng tr\u00f2n ti\u1ebfp x\u00fac trong th\u00ec: OO\u2032 = R \u2212 R\u2032 (Ti\u1ebfp \u0111i\u1ec3m n\u1eb1m tr\u00ean \u0111o\u1ea1n n\u1ed1i t\u00e2m). N\u1ebfu hai \u0111\u01b0\u1eddng tr\u00f2n \u0111\u1ef1ng nhau th\u00ec: OO\u2032 < R \u2212 R\u2032. 76 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M II. B\u00e0i t\u1eadp: B\u00e0i 1. Cho (O; 4cm) v\u00e0 (O\u2032; 6cm) ; OO\u2032 = 12cm. Ch\u1ee9ng minh r\u1eb1ng (O) v\u00e0 (O\u2032) \u1edf ngo\u00e0i nhau. B\u00e0i 2. Cho \u0111i\u1ec3m A thu\u1ed9c \u0111o\u1ea1n th\u1eb3ng OO\u2032. V\u1ebd (O; OA) v\u00e0 (O; O\u2032A). Ch\u1ee9ng minh r\u1eb1ng (O) v\u00e0 (O\u2032) ti\u1ebfp x\u00fac ngo\u00e0i. B\u00e0i 3. Cho (O; 12cm) v\u00e0 (O\u2032; 5cm) , gi\u1ea3 s\u1eed OO\u2032 = 13cm. Ch\u1ee9ng minh r\u1eb1ng (O) v\u00e0 (O\u2032) c\u1eaft nhau. B\u00e0i 4. Cho \u0111i\u1ec3m A thu\u1ed9c (O) . G\u1ecdi O\u2032 l\u00e0 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1eddng k\u00ednh OA. Ch\u1ee9ng minh r\u1eb1ng (O) v\u00e0 (O\u2032) ti\u1ebfp x\u00fac trong. | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 77","| C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT | Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m B\u00e0i 5. Cho \u0111o\u1ea1n OO\u2032 = 3cm. V\u1ebd (O; 2cm) v\u00e0 (O\u2032; 7cm) . Ch\u1ee9ng minh r\u1eb1ng (O) v\u00e0 (O\u2032) \u0111\u1ef1ng nhau. B\u00e0i 6. Cho (O; 2cm) v\u00e0 (O\u2032; 1cm) ; OO\u2032 = 6cm. H\u00e3y cho bi\u1ebft v\u1ecb tr\u00ed gi\u1eefa hai \u0111\u01b0\u1eddng tr\u00f2n. B\u00e0i 7. Cho (O; 3cm) v\u00e0 (O\u2032; 1cm) ; OO\u2032 = 4cm. H\u00e3y cho bi\u1ebft v\u1ecb tr\u00ed gi\u1eefa hai \u0111\u01b0\u1eddng tr\u00f2n. B\u00e0i 8. Cho (O; 13cm) v\u00e0 (O; 15cm) c\u1eaft nhau t\u1ea1i hai \u0111i\u1ec3m M v\u00e0 N. Bi\u1ebft M N = 24cm. H\u00e3y t\u00ednh \u0111o\u1ea1n n\u1ed1i t\u00e2m OO\u2032. B\u00e0i 9. Cho hai \u0111\u01b0\u1eddng tr\u00f2n \u0111\u1ed3ng t\u00e2m (O; R) v\u00e0 (O; R) . T\u00ednh b\u00e1n k\u00ednh r c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n ti\u1ebfp x\u00fac v\u1edbi c\u1ea3 hai \u0111\u01b0\u1eddng tr\u00f2n tr\u00ean bi\u1ebft R = 12cm, R\u2032 = 8cm. B\u00e0i 10. Cho hai \u0111\u01b0\u1eddng tr\u00f2n (O) v\u00e0 (I) ti\u1ebfp x\u00fac ngo\u00e0i t\u1ea1i A. G\u1ecdi DE l\u00e0 ti\u1ebfp tuy\u1ebfn chung ngo\u00e0i c\u1ee7a hai \u0111\u01b0\u1eddng tr\u00f2n (D thu\u1ed9c (O) , E thu\u1ed9c (I)). Ti\u1ebfp tuy\u1ebfn chung trong t\u1ea1i A c\u1ee7a hai \u0111\u01b0\u1eddng tr\u00f2n c\u1eaft ti\u1ebfp tuy\u1ebfn chung ngo\u00e0i DE t\u1ea1i M. Ch\u1ee9ng minh r\u1eb1ng tam gi\u00e1c DEA vu\u00f4ng. B\u00e0i 11. Cho (O) ti\u1ebfp x\u00fac ngo\u00e0i v\u1edbi (O\u2032) t\u1ea1i \u0111i\u1ec3m M. V\u1ebd ti\u1ebfp tuy\u1ebfn chung ngo\u00e0i v\u1edbi hai \u0111\u01b0\u1eddng tr\u00f2n (O) v\u00e0 (O) c\u00f3 ti\u1ebfp \u0111i\u1ec3m l\u1ea7n l\u01b0\u1ee3t l\u00e0 A v\u00e0 B. Ch\u1ee9ng minh t\u1eb1ng tam gi\u00e1c AM B vu\u00f4ng. B\u00e0i 12. Cho (O; 12cm) v\u00e0 (O\u2032; 5cm) ; OO\u2032 = 13cm. 1) Ch\u1ee9ng minh r\u1eb1ng OO\u2032 l\u00e0 \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a AB, H l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB; 2) T\u00ednh AH; OH; O\u2032H; OO\u2032. B\u00e0i 13. Cho \u0111i\u1ec3m A thu\u1ed9c (O) . G\u1ecdi O\u2032 l\u00e0 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1eddng k\u00ednh OA. 1) H\u00e3y cho bi\u1ebft v\u1ecb tr\u00ed c\u1ee7a hai \u0111\u01b0\u1eddng tr\u00f2n; 2) D\u00e2y AB c\u1ee7a (O) c\u1eaft (O) t\u1ea1i C. Tam gi\u00e1c AO\u2032C l\u00e0 tam gi\u00e1c g\u00ec?; 3) Ch\u1ee9ng minh r\u1eb1ng O\u2032C\/\/OB; 4) Ch\u1ee9ng minh C l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB. B\u00e0i 14. Cho (O) v\u00e0 (O\u2032) ti\u1ebfp x\u00fac ngo\u00e0i t\u1ea1i A. DE l\u00e0 ti\u1ebfp tuy\u1ebfn chung ngo\u00e0i v\u1edbi ti\u1ebfp \u0111i\u1ec3m D c\u1ee7a (O) v\u00e0 ti\u1ebfp \u0111i\u1ec3m E c\u1ee7a (O\u2032) . Ti\u1ebfp tuy\u1ebfn chung trong t\u1ea1i A c\u1eaft DE t\u1ea1i I. 1) Ch\u1ee9ng minh I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a DE v\u00e0 DAE vu\u00f4ng t\u1ea1i A; 2) K\u1ebb \u0111\u01b0\u1eddng k\u00ednh AB c\u1ee7a (O) , \u0111\u01b0\u1eddng k\u00ednh AC c\u1ee7a (O\u2032) , BD c\u1eaft CE t\u1ea1i M. Ch\u1ee9ng minh r\u1eb1ng c\u00e1c tam gi\u00e1c: ABD v\u00e0 ACE l\u00e0 c\u00e1c tam gi\u00e1c vu\u00f4ng. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 78 N\u0103m h\u1ecdc: 2023 - 2024","Ch\u01b0\u01a1ng 3 G\u00d3C V\u1edaI \u0110\u01af\u1edcNG TR\u00d2N B\u00e0i 1 G\u00f3c v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M Trong m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n: G\u00f3c \u1edf t\u00e2m: L\u00e0 g\u00f3c t\u1ea1o b\u1edfi hai b\u00e1n k\u00ednh. G\u00f3c n\u1ed9i ti\u1ebfp: L\u00e0 g\u00f3c c\u00f3 \u0111\u1ec9nh thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n, hai c\u1ea1nh l\u00e0 hai d\u00e2y cung. S\u1ed1 \u0111o cung tr\u00f2n: L\u00e0 s\u1ed1 \u0111o g\u00f3c \u1edf t\u00e2m ch\u1eafn cung \u0111\u00f3. 1. Li\u00ean h\u1ec7 gi\u1eefa g\u00f3c v\u00e0 cung tr\u00f2n: G\u00f3c \u1edf t\u00e2m b\u1eb1ng s\u1ed1 \u0111o cung b\u1ecb ch\u1eb5n. G\u00f3c n\u1ed9i ti\u1ebfp b\u1eb1ng n\u1eeda s\u1ed1 \u0111o cung b\u1ecb ch\u1eafn. G\u00f3c t\u1ea1o b\u1edfi ti\u1ebfp tuy\u1ebfn v\u00e0 d\u00e2y cung b\u1eb1ng n\u1eeda s\u1ed1 \u0111o cung b\u1ecb ch\u1eb5n. G\u00f3c n\u1ed9i ti\u1ebfp ch\u1eafn n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n b\u1eb1ng 90\u25e6. 2. Li\u00ean h\u1ec7 gi\u1eefa c\u00e1c g\u00f3c: G\u00f3c n\u1ed9i ti\u1ebfp b\u1eb1ng n\u1eeda g\u00f3c \u1edf t\u00e2m c\u00f9ng ch\u1eafn m\u1ed9t cung. Hai n\u1ed9i ti\u1ebfp c\u00f9ng ch\u1eafn m\u1ed9t cung ho\u1eb7c hai cung b\u1eb1ng nhau th\u00ec b\u1eb1ng nhau. G\u00f3c t\u1ea1o b\u1edfi ti\u1ebfp tuy\u1ebfn v\u00e0 d\u00e2y cung b\u1eb1ng g\u00f3c n\u1ed9i ti\u1ebfp c\u00f9ng ch\u1eb5n m\u1ed9t cung. 3. G\u00f3c c\u00f3 \u0111\u1ec9nh b\u00ean trong \u0111\u01b0\u1eddng tr\u00f2n v\u00e0 g\u00f3c c\u00f3 \u0111\u1ec9nh b\u00ean ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n: G\u00f3c c\u00f3 \u0111\u1ec9nh b\u00ean trong \u0111\u01b0\u1eddng tr\u00f2n b\u1eb1ng n\u1eeda t\u1ed5ng s\u1ed1 \u0111o hai cung b\u1ecb ch\u1eafn. G\u00f3c c\u00f3 \u0111\u1ec9nh b\u00ean ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n b\u1eb1ng n\u1eeda hi\u1ec7u s\u1ed1 \u0111o hai cung b\u1ecb ch\u1eafn. II. B\u00e0i t\u1eadp: G\u00d3C \u1ede T\u00c2M, G\u00d3C N\u1ed8I TI\u1ebeP | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 79","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m B\u00e0i 1. Tr\u00ean (O) l\u1ea5y hai \u0111i\u1ec3m A v\u00e0 B sao cho AOB = 60\u25e6. T\u00ednh s\u0111 \u2322 nh\u1ecf. AB | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT B\u00e0i 2. Tr\u00ean (O) l\u1ea5y hai \u0111i\u1ec3m C v\u00e0 D sao cho \u2322 = 45\u25e6. T\u00ednh s\u1ed1 \u0111o COD. s\u0111C D B\u00e0i 3. Cho \u0111\u01b0\u1eddng tr\u00f2n (O) , v\u1ebd g\u00f3c n\u1ed9i ti\u1ebfp BAC = 30\u25e6, (B, C thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O). T\u00ednh s\u1ed1 \u0111o cung BC l\u1edbn. B\u00e0i 4. Cho (O) \u0111\u01b0\u1eddng k\u00ednh AC, l\u1ea5y B tr\u00ean (O) . Ch\u1ee9ng minh r\u1eb1ng BOC = 2BAC. 80 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 B\u00e0i 5. Cho (O) \u0111\u01b0\u1eddng k\u00ednh AC, l\u1ea5y B v\u00e0 E tr\u00ean (O) sao cho B v\u00e0 E n\u1eb1m v\u1ec1 hai ph\u00eda \u0111\u01b0\u1eddng k\u00ednh AC. Ch\u1ee9ng minh r\u1eb1ng BOE = 2BAE. B\u00e0i 6. Cho \u0111\u01b0\u1eddng tr\u00f2n (O) , v\u1ebd hai d\u00e2y BA v\u00e0 BC sao cho ABC = 25\u25e6. T\u00ednh s\u1ed1 \u0111o cung nh\u1ecf AC v\u00e0 s\u1ed1 \u0111o cung l\u1edbn AC. B\u00e0i 7. Tr\u00ean (O) l\u1ea5y hai \u0111i\u1ec3m A v\u00e0 B sao cho g\u00f3c \u1edf t\u00e2m AOB = 70\u25e6. L\u1ea5y \u0111i\u1ec3m C b\u1ea5t k\u1ef3 tr\u00ean \u2322\u2322 AB l\u1edbn. T\u00ednh s\u0111AB nh\u1ecf v\u00e0 s\u1ed1 \u0111o ACB. B\u00e0i 8. Tr\u00ean (O) l\u1ea5y ba \u0111i\u1ec3m A, B v\u00e0 C sao cho g\u00f3c n\u1ed9i ti\u1ebfp C AB = 40\u25e6. T\u00ednh \u2322 nh\u1ecf v\u00e0 s\u1ed1 sC B \u0111o COB. B\u00e0i 9. Tr\u00ean (O; R) v\u1ebd d\u00e2y AB = R. T\u00ednh s\u1ed1 \u0111o cung l\u1edbn AB. | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M B\u00e0i 10. V\u1ebd g\u00f3c BAC n\u1ed9i ti\u1ebfp (O) , bi\u1ebft g\u00f3c BAC = 45\u25e6. Ch\u1ee9ng minh r\u1eb1ng Hai b\u00e1n k\u00ednh OB v\u00e0 OC vu\u00f4ng g\u00f3c nhau. B\u00e0i 11. Tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O l\u1ea5y ba \u0111i\u1ec3m D, E, F theo th\u1ee9 t\u1ef1 \u0111\u00f3 sao cho EDF = 30\u25e6. Ch\u1ee9ng minh r\u1eb1ng \u25b3OEF \u0111\u1ec1u. B\u00e0i 12. Tr\u00ean (O) l\u1ea5y hai \u0111i\u1ec3m A v\u00e0 B sao cho AOB vu\u00f4ng t\u1ea1i O. L\u1ea5y \u0111i\u1ec3m C tr\u00ean cung nh\u1ecf AB, g\u1ecdi Cx l\u00e0 tia \u0111\u1ed1i c\u1ee7a tia CA. Ch\u1ee9ng minh r\u1eb1ng: 1) ABC + BAC = 45\u25e6; 2) T\u00ednh BCx. B\u00e0i 13. Tr\u00ean (O) l\u1ea5y hai \u0111i\u1ec3m M v\u00e0 N sao cho M ON = 120\u25e6. L\u1ea5y \u0111i\u1ec3m E tr\u00ean cung nh\u1ecf M N, g\u1ecdi Ex l\u00e0 tia \u0111\u1ed1i c\u1ee7a tia EM. T\u00ednh N Ex. B\u00e0i 14. Cho (O) c\u00f3 hai d\u00e2y cung AC v\u00e0 BD c\u1eaft nhau t\u1ea1i I. Ch\u1ee9ng minh r\u1eb1ng: 1) CAD = CBD;S 2) \u25b3AIB \u25b3DIC. B\u00e0i 15. Cho (O) c\u00f3 d\u00e2y cung AC v\u00e0 \u0111\u01b0\u1eddng k\u00ednh BD c\u1eaft nhau t\u1ea1i I. Ch\u1ee9ng minh r\u1eb1ng: 1) IA.IC = IB.ID; 2) IA.IC = R2 \u2212 OI2. | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 81","| C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT | Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m SS B\u00e0i 16. L\u1ea5y 4 \u0111i\u1ec3m A, B, C, D theo th\u1ee9 t\u1ef1 \u0111\u00f3 tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n (O) sao cho hai \u0111\u01b0\u1eddng th\u1eb3ng AB v\u00e0 CD c\u1eaft nhau t\u1ea1i m\u1ed9t \u0111i\u1ec3m I b\u00ean ngo\u00e0i (O) . Ch\u1ee9ng minh r\u1eb1ng: 1) BAC = BDC; 2) \u25b3IBD \u25b3ICA; 3) \u25b3IBC \u25b3IDA. G\u00d3C T\u1ea0O B\u1edeI TI\u1ebeP TUY\u1ebeN V\u00c0 D\u00c2Y CUNG B\u00e0i 1. Tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O l\u1ea5y hai \u0111i\u1ec3m A, B b\u1ea5t k\u00ec sao cho s\u1ed1 \u0111o cung AB l\u00e0 80\u25e6. V\u1ebd ti\u1ebfp tuy\u1ebfn Ax. T\u00ednh BAx. B\u00e0i 2. Tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O l\u1ea5y hai \u0111i\u1ec3m B, A. T\u1eeb A v\u1ebd ti\u1ebfp tuy\u1ebfn Ax c\u1ee7a (O) sao cho xAB = 70\u25e6. L\u1ea5y C b\u1ea5t k\u00ec. T\u00ednh ACB. B\u00e0i 3. Cho (O) c\u00f3 AOB = 120\u25e6. T\u1eeb A v\u1ebd ti\u1ebfp tuy\u1ebfn Ax c\u1ee7a (O) , l\u1ea5y C b\u1ea5t k\u00ec. T\u00ednh ACB v\u00e0 xAB. 82 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 B\u00e0i 4. Cho n\u1eeda (O) \u0111\u01b0\u1eddng k\u00ednh AC, g\u1ecdi Ax l\u00e0 ti\u1ebfp tuy\u1ebfn c\u1ee7a n\u1eeda (O) t\u1ea1i A (Ax c\u00f9ng ph\u00eda v\u1edbi n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n), AB l\u00e0 d\u00e2y cung c\u1ee7a (O) . Ch\u1ee9ng minh r\u1eb1ng xAB = ACB. B\u00e0i 5. Cho tam gi\u00e1c BAC c\u00e2n t\u1ea1i A n\u1ed9i ti\u1ebfp (O) . L\u1ea5y \u0111i\u1ec3m D thu\u1ed9c cung BC kh\u00f4ng ch\u1ee9a A.SS | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M Ch\u1ee9ng minh r\u1eb1ng: S 1) ADC = ABC; S 2) ADC = ACB. B\u00e0i 6. Cho AB l\u00e0 \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a (O; R) . V\u1ebd hai d\u00e2y cung AD v\u00e0 BC c\u1eaft nhau t\u1ea1i E. T\u1eeb E k\u1ebb EF vu\u00f4ng g\u00f3c v\u1edbi AB t\u1ea1i F. Ch\u1ee9ng minh r\u1eb1ng: 1) ADB = ACB = 90\u25e6; 2) \u25b3AEF \u25b3ADB; 3) \u25b3BEF \u25b3BAC. B\u00e0i 7. Cho tam gi\u00e1c ABC nh\u1ecdn c\u00f3 AD l\u00e0 \u0111\u01b0\u1eddng cao. \u0110\u01b0\u1eddng tr\u00f2n t\u00e2m O \u0111\u01b0\u1eddng k\u00ednh BC c\u1eaft AB v\u00e0 AC l\u1ea7n l\u01b0\u1ee3t t\u1ea1i F v\u00e0 E. 1) T\u00ednh s\u1ed1 \u0111o BEC v\u00e0 CF B; 2) Ch\u1ee9ng minh AD, BE, CF \u0111\u1ed3ng quy (c\u1eaft nhau t\u1ea1i m\u1ed9t \u0111i\u1ec3m). B\u00e0i 8. Cho \u25b3ABC nh\u1ecdn n\u1ed9i ti\u1ebfp (O) k\u1ebb \u0111\u01b0\u1eddng k\u00ednh AD, AH l\u00e0 \u0111\u01b0\u1eddng cao c\u1ee7a \u25b3ABC. 1) T\u00ednh ACD; 2) Ch\u1ee9ng minh r\u1eb1ng ABH = ADC; 3) Ch\u1ee9ng minh r\u1eb1ng \u25b3ABH \u25b3ADC. B\u00e0i 9. L\u1ea5y M thu\u1ed9c n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1eddng k\u00ednh AB. T\u1eeb A k\u1ebb ti\u1ebfp tuy\u1ebfn Ax c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n O, M H vu\u00f4ng g\u00f3c v\u1edbi Ax t\u1ea1i H. Ch\u1ee9ng minh r\u1eb1ng: 1) M AH = M BA; 2) \u25b3M BE \u25b3M BA; 3) M H.AB = M A2. B\u00e0i 10. Cho AB l\u00e0 d\u00e2y cung c\u1ee7a (O) . L\u1ea5y \u0111i\u1ec3m D tr\u00ean tia \u0111\u1ed1i c\u1ee7a tia BA. G\u1ecdi C l\u00e0 \u0111i\u1ec3m ch\u00ednh gi\u1eefa c\u1ee7a cung AB l\u1edbn. CD c\u1eaft (O) t\u1ea1i E. Ch\u1ee9ng minh r\u1eb1ng: | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 83","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m S 1) CAB = CEA; 2) \u25b3CAE \u25b3CAD v\u00e0 suy ra CA2 = CE.CD. B\u00e0i 11. T\u1eeb \u0111i\u1ec3m M n\u1eb1m ngo\u00e0i (O) k\u1ebb c\u00e1c tuy\u1ebfn M BA (B n\u1eb1m gi\u1eefa A v\u00e0 M ), b\u00e1n k\u00ednh OC vu\u00f4ng g\u00f3c v\u1edbi d\u00e2y cung AB, v\u1edbi C thu\u1ed9c cung AB l\u1edbn. cm c\u1eaft (O) t\u1ea1i K. Ch\u1ee9ng minh r\u1eb1ng: 1) C l\u00e0 \u0111i\u1ec3m ch\u00ednh gi\u1eefa c\u1ee7a cung AB l\u1edbn; | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT 2) AKC = CAB; 3) CA2 = CK.CM. B\u00e0i 12. Cho (O) c\u00f3 \u0111\u01b0\u1eddng k\u00ednh AB v\u00e0 \u0111i\u1ec3m I thu\u1ed9c tia \u0111\u1ed1i tia BA, t\u1eeb I v\u1ebd ti\u1ebfp tuy\u1ebfn IT v\u1edbi (O) . Ch\u1ee9ng minh r\u1eb1ng: 1) BT I = T AB; 2) \u25b3IT B \u25b3IAT.SS B\u00e0i 13. T\u1eeb \u0111i\u1ec3m A b\u00ean ngo\u00e0i (O) , v\u1ebd c\u00e1t tuy\u1ebfn ABC v\u00e0 ti\u1ebfp tuy\u1ebfn AT v\u1edbi (O) . Ch\u1ee9ng minh r\u1eb1ng: 1) \u25b3AT B \u25b3ACT ; 2) AT 2 = AB.AC. B\u00e0i 14. Cho \u25b3ABC nh\u1ecdn n\u1ed9i ti\u1ebfp (O) , v\u1ebd ED vu\u00f4ng g\u00f3c v\u1edbi \u0111o\u1ea1n OC t\u1ea1i I (I kh\u00f4ng tr\u00f9ng v\u1edbi C v\u00e0 O, E thu\u1ed9c CB, D thu\u1ed9c CA) v\u1ebd ti\u1ebfp tuy\u1ebfn xy v\u1edbi (O) t\u1ea1i C. Ch\u1ee9ng minh r\u1eb1ng: 1) \u25b3ABC \u25b3CDE; S 2) CE.CB = CD.CA. G\u00d3C C\u00d3 \u0110\u1ec8NH B\u00caN TRONG, B\u00caN NGO\u00c0I \u0110\u01af\u1edcNG TR\u00d2N B\u00e0i 1. Cho hai d\u00e2y cung AC v\u00e0 BD c\u1ee7a (O) c\u1eaft nhau t\u1ea1i E b\u00ean trong \u0111\u01b0\u1eddng tr\u00f2n. Ch\u1ee9ng minh r\u1eb1ng: 1) BEC = ECD + EDC; 1\u2322 \u2322 2) BEC = s\u0111BC + s\u0111AD . 2 (T\u1eeb \u0111\u00f3 ph\u00e1t bi\u1ec3u m\u1ed1i li\u00ean h\u1ec7 gi\u1eefa g\u00f3c c\u00f3 \u0111\u1ec9nh b\u00ean trong \u0111\u01b0\u1eddng v\u00e0 cung b\u1ecb ch\u1eafn) B\u00e0i 2. T\u1eeb \u0111i\u1ec3m E b\u00ean ngo\u00e0i (O) d\u1ef1ng hai c\u00e1t tuy\u1ebfn EBA v\u00e0 ECD v\u1edbi (O) . Ch\u1ee9ng minh r\u1eb1ng: 1) AED = ACD \u2212 CAB; 1 \u2322\u2322 . 2) AED = 2 s\u0111AD \u2212 s\u0111BC (T\u1eeb \u0111\u00f3 ph\u00e1t bi\u1ec3u m\u1ed1i li\u00ean h\u1ec7 gi\u1eefa g\u00f3c c\u00f3 \u0111\u1ec9nh b\u00ean ngo\u00e0i \u0111\u01b0\u1eddng v\u00e0 cung b\u1ecb ch\u1eafn) B\u00e0i 3. Cho \u0111\u01b0\u1eddng tr\u00f2n (O) c\u00f3 \u0111\u01b0\u1eddng k\u00ednh AB, l\u1ea5y hai \u0111i\u1ec3m D v\u00e0 C sao cho hai d\u00e2y cung DB v\u00e0 \u2322\u2322 AC c\u1eaft nhau t\u1ea1i m\u1ed9t \u0111i\u1ec3m E n\u1eb1m trong \u0111\u01b0\u1eddng tr\u00f2n. Bi\u1ebft s\u0111AD = 60\u25e6 v\u00e0 s\u0111BC = 100\u25e6. T\u00ednh CEB. 84 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 B\u00e0i 4. Cho \u0111\u01b0\u1eddng tr\u00f2n (O) c\u00f3 \u0111\u01b0\u1eddng k\u00ednh AB, l\u1ea5y hai \u0111i\u1ec3m D v\u00e0 C sao cho hai d\u00e2y cung DB v\u00e0 AC c\u1eaft nhau t\u1ea1i m\u1ed9t \u0111i\u1ec3m E n\u1eb1m trong \u0111\u01b0\u1eddng tr\u00f2n. Bi\u1ebft CEB = 100\u25e6 v\u00e0 \u2322 = 86\u25e6. T\u00ednh s\u0111BC \u2322 s\u0111AD. B\u00e0i 5. Cho \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m (O) . T\u1eeb \u0111i\u1ec3m I n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n v\u1ebd 2 c\u00e1t tuy\u1ebfn IDC v\u00e0 IAB | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M \u2322\u2322 bi\u1ebft s\u0111BC = 100\u25e6 v\u00e0 s\u0111AD = 40\u25e6. T\u00ednh BIC. B\u00e0i 6. T\u1eeb \u0111i\u1ec3m E b\u00ean ngo\u00e0i (O) d\u1ef1ng hai c\u00e1t tuy\u1ebfn EDA v\u00e0 ECB v\u1edbi (O) sao cho DOC = 40\u25e6 v\u00e0 AOB = 110\u25e6. T\u00ednh s\u1ed1 \u0111o AEB. B\u00e0i 7. Tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n (O) l\u1ea5y 4 \u0111i\u1ec3m A, B, C, D theo th\u1ee9 t\u1ef1 \u0111\u00f3 sao cho BOC = AOD = 90\u25e6. Ch\u1ee9ng minh r\u1eb1ng BD\u22a5AC. B\u00e0i 8. Tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n (O) l\u1ea5y b\u1ed1n \u0111i\u1ec3m F, H, I, J theo th\u1ee9 t\u1ef1 \u0111\u00f3 sao cho HOF = 60\u25e6 v\u00e0 IOJ = 120\u25e6. Ch\u1ee9ng minh r\u1eb1ng F I\u22a5HJ. B\u00e0i 9. Cho \u25b3ABC c\u00e2n t\u1ea1i A n\u1ed9i ti\u1ebfp (O) c\u00f3 BC l\u00e0 \u0111\u01b0\u1eddng k\u00ednh. Tr\u00ean n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n kh\u00f4ng ch\u1ee9a A l\u1ea5y \u0111i\u1ec3m D. AD c\u1eaft CB t\u1ea1i M. Ch\u1ee9ng minh r\u1eb1ng: 1) ADC = ACB; 2) AM.AD = AC2. B\u00e0i 10. Cho tam gi\u00e1c ABC c\u00e2n t\u1ea1i A n\u1ed9i ti\u1ebfp (O) . L\u1ea5y D tr\u00ean cung BC nh\u1ecf, d\u00e2y AD c\u1eaft d\u00e2y BC t\u1ea1i I. Ch\u1ee9ng minh r\u1eb1ng: 1) \u25b3AIC \u25b3ADC; S 2) AI.AD = AC2. B\u00e0i 11. Cho \u25b3ABC c\u00e2n t\u1ea1i A n\u1ed9i ti\u1ebfp (O) . Tr\u00ean n\u1eefa \u0111\u01b0\u1eddng tr\u00f2n kh\u00f4ng ch\u1ee9a A l\u1ea5y hai \u0111i\u1ec3m D v\u00e0 E. AD v\u00e0 AE c\u1eaft CB t\u1ea1i M v\u00e0 N. Ch\u1ee9ng minh r\u1eb1ng AM.AD = AN.AE. B\u00e0i 12. Cho \u25b3ABC c\u00e2n t\u1ea1i A n\u1ed9i ti\u1ebfp (O) . Tr\u00ean cung nh\u1ecf AB l\u1ea5y M, AM c\u1eaft tia CB t\u1ea1i N. Ch\u1ee9ng minh r\u1eb1ng: 1) M BA = AN C; 2) AM.AN = AB2. B\u00e0i 13. Cho n\u1eeda (O) \u0111\u01b0\u1eddng k\u00ednh AB, l\u1ea5y \u0111i\u1ec3m E tr\u00ean n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n, g\u1ecdi C v\u00e0 D l\u1ea7n l\u01b0\u1ee3t l\u00e0 \u0111i\u1ec3m ch\u00ednh gi\u1eefa c\u1ee7a c\u00e1c cung AE v\u00e0 cung BE. G\u1ecdi M v\u00e0 N l\u1ea7n l\u01b0\u1ee3t l\u00e0 giao \u0111i\u1ec3m c\u1ee7a CD v\u1edbi c\u00e1c d\u00e2y cung AE v\u00e0 BE. Ch\u1ee9ng minh \u25b3EM N vu\u00f4ng c\u00e2n. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 2 Luy\u1ec7n t\u1eadp: G\u00f3c v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 85","| C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT | Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m \u221a B\u00e0i 1. Cho \u0111\u01b0\u1eddng tr\u00f2n (O; R = 1cm) c\u00f3 d\u00e2y cung AB b\u1eb1ng 2, E thu\u1ed9c cung l\u1edbn AB. T\u00ednh: 1) AOB; 2) AEB. B\u00e0i 2. Cho \u0111\u01b0\u1eddng tr\u00f2n (O; R) c\u00f3 d\u00e2y cung AB b\u1eb1ng R. Tr\u00ean cung l\u1edbn AB l\u1ea5y \u0111i\u1ec3m C. 1) Tam gi\u00e1c AOB l\u00e0 tam gi\u00e1c g\u00ec?; 2) T\u00ednh ACB. \u221a B\u00e0i 3. Cho \u0111\u01b0\u1eddng tr\u00f2n (O; R) c\u00f3 d\u00e2y cung AB b\u1eb1ng R 3, I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB. Tr\u00ean cung l\u1edbn AB l\u1ea5y \u0111i\u1ec3m C. T\u00ednh: 1) AOI; 2) AOB; 3) ACB. B\u00e0i 4. Cho tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp (O) . V\u1ebd \u0111\u01b0\u1eddng k\u00ednh AD c\u1ee7a (O) , \u0111\u01b0\u1eddng cao AH c\u1ee7a tam gi\u00e1c ABC c\u1eaft (O) t\u1ea1i E. Ch\u1ee9ng minh r\u1eb1ng: 1) ED\/\/BC; 2) BEDC l\u00e0 h\u00ecnh thang c\u00e2n; 3) BAE = DAC. B\u00e0i 5. Cho (O) c\u00f3 hai d\u00e2y AB, CD song song nhau t\u1ea1o th\u00e0nh h\u00ecnh thang ABCD n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n. Ch\u1ee9ng minh r\u1eb1ng: 1) Hai cung AD v\u00e0 BC b\u1eb1ng nhau; \u2322\u2322 2) s\u0111DB = s\u0111AC; 3) ADC = BCD. B\u00e0i 6. Tr\u00ean (O) l\u1ea5y 4 \u0111i\u1ec3m A, B, C, D theo th\u1ee9 t\u1ef1 \u0111\u00f3 sao cho hai d\u00e2y AB v\u00e0 CD b\u1eb1ng nhau v\u00e0 kh\u00f4ng c\u1eaft nhau. Ch\u1ee9ng minh r\u1eb1ng: 1) AD\/\/BC; 2) CBD = ADB; 3) ABCD l\u00e0 h\u00ecnh thang c\u00e2n. B\u00e0i 7. Tr\u00ean (O) l\u1ea5y 4 \u0111i\u1ec3m A, C, B, D theo th\u1ee9 t\u1ef1 \u0111\u00f3 sao cho hai d\u00e2y cung AB v\u00e0 CD b\u1eb1ng nhau v\u00e0 c\u1eaft nhau t\u1ea1i I. Ch\u1ee9ng minh r\u1eb1ng: \u2322\u2322 1) s\u0111CB = s\u0111AD; 2) AC\/\/BD; 3) ACBD l\u00e0 h\u00ecnh thang c\u00e2n. 86 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M B\u00e0i 8. Cho tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp (O) c\u00f3 hai \u0111\u01b0\u1eddng cao AD v\u00e0 BE c\u1eaft nhau t\u1ea1i H, tia AH c\u1eaft (O) t\u1ea1i M, tia BH c\u1eaft (O) t\u1ea1i F. Ch\u1ee9ng minh r\u1eb1ng: 1) Tam gi\u00e1c Hcm c\u00e2n t\u1ea1i C; 2) CM = CF . B\u00e0i 9. Cho tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp (O) , v\u1ebd b\u00e1n k\u00ednh OI vu\u00f4ng g\u00f3c d\u00e2y BC, g\u1ecdi D l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AI v\u00e0 BC. Ch\u1ee9ng minh r\u1eb1ng: 1) AI l\u00e0 tia ph\u00e2n gi\u00e1c BAC; 2) ID.IA = IC2. \u221a B\u00e0i 10. Cho (O; R) c\u00f3 d\u00e2y cung AB b\u1eb1ng R 3, l\u1ea5y C thu\u1ed9c cung l\u1edbn AB, v\u1ebd \u0111\u01b0\u1eddng k\u00ednh AD. T\u00ednh s\u1ed1 \u0111o g\u00f3c: 1) ADB; 2) ACB. B\u00e0i 11. Cho \u0111\u01b0\u1eddng tr\u00f2n (O; R) c\u00f3 \u0111\u01b0\u1eddng k\u00ednh AB. G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m OB, v\u1ebd d\u00e2y cung CD vu\u00f4ng g\u00f3c v\u1edbi AB t\u1ea1i I. 1) Ch\u1ee9ng minh r\u1eb1ng tam gi\u00e1c OBC \u0111\u1ec1u; 2) T\u00ednh s\u1ed1 \u0111o cung nh\u1ecf DC; 3) T\u00ednh s\u1ed1 \u0111o DAC; 4) Ch\u1ee9ng minh r\u1eb1ng \u25b3ACD \u0111\u1ec1u. B\u00e0i 12. Cho tam gi\u00e1c ABC nh\u1ecdn n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n (O) . V\u1ebd \u0111\u01b0\u1eddng cao AH c\u1ee7a tam gi\u00e1c v\u00e0 \u0111\u01b0\u1eddng k\u00ednh AD c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n. Ch\u1ee9ng minh r\u1eb1ng BAH = DAC. B\u00e0i 13. Cho tam gi\u00e1c ABC c\u00e2n t\u1ea1i A n\u1ed9i ti\u1ebfp (O) . L\u1ea5y D tr\u00ean cung BC, d\u00e2y AD c\u1eaft d\u00e2y BC t\u1ea1i I. Ch\u1ee9ng minh r\u1eb1ng: 1) ACB = ADC; 2) AI.AD = AC2. B\u00e0i 14. Cho tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n (O; R) , v\u1ebd \u0111\u01b0\u1eddng k\u00ednh BD. Ch\u1ee9ng minh \u0111\u1ecbnh abc l\u00fd = = = 2R. sin A sin B sin C B\u00e0i 15. Cho tam gi\u00e1c ABC nh\u1ecdn n\u1ed9i ti\u1ebfp (O) . Trong tam gi\u00e1c ABC v\u1ebd hai \u0111\u01b0\u1eddng cao AD v\u00e0 BE c\u1eaft nhau t\u1ea1i H. AH k\u00e9o d\u00e0i c\u1eaft (O) t\u1ea1i F. Ch\u1ee9ng minh r\u1eb1ng: 1) EBC = DAC; 2) \u25b3BF H c\u00e2n; 3) D l\u00e0 trung \u0111i\u1ec3m c\u1ee7a F H. | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 87","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m B\u00e0i 16. Cho (O1; R) v\u00e0 (O2; R) c\u1eaft nhau t\u1ea1i hai \u0111i\u1ec3m A v\u00e0 B. M\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua A c\u1eaft O1 t\u1ea1i C v\u00e0 c\u1eaft O2 t\u1ea1i D. Ch\u1ee9ng minh r\u1eb1ng: 1) T\u1ee9 gi\u00e1c AO1BO2 l\u00e0 h\u00ecnh thoi; 2) Tam gi\u00e1c BCD c\u00e2n t\u1ea1i B. B\u00e0i 17. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A, v\u1ebd c\u00e1c \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m C b\u00e1n k\u00ednh AC v\u00e0 \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m B b\u00e1n k\u00ednh AB, hai \u0111\u01b0\u1eddng tr\u00f2n n\u00e0y c\u1eaft nhau t\u1ea1i \u0111i\u1ec3m th\u1ee9 hai l\u00e0 D. Qua A v\u1ebd m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng c\u1eaft hai \u0111\u01b0\u1eddng tr\u00f2n tr\u00ean t\u1ea1i E v\u00e0 F. Ch\u1ee9ng minh r\u1eb1ng: | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT S 1) AF D = ACB; 2) \u25b3DEF vu\u00f4ng t\u1ea1i D. B\u00e0i 18. Cho h\u00ecnh vu\u00f4ng ABCD, l\u1ea5y \u0111i\u1ec3m I tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m C b\u00e1n k\u00ednh CB sao cho BI c\u1eaft \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m A b\u00e1n k\u00ednh BA t\u1ea1i \u0111i\u1ec3m J (J n\u1eb1m gi\u1eefa A v\u00e0 B). 1) T\u00ednh DIB; 2) Ch\u1ee9ng minh r\u1eb1ng DJI = DBJ + JDB; 3) Ch\u1ee9ng minh r\u1eb1ng \u25b3DIJ vu\u00f4ng c\u00e2n t\u1ea1i D. B\u00e0i 19. Cho hai \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m A v\u00e0 B ti\u1ebfp x\u00fac ngo\u00e0i nhau t\u1ea1i \u0111i\u1ec3m C, v\u1ebd ti\u1ebfp tuy\u1ebfn chung Cx v\u1edbi hai \u0111\u01b0\u1eddng tr\u00f2n t\u1ea1i C. Tr\u00ean Cx l\u1ea5y D b\u1ea5t k\u1ef3, t\u1eeb D v\u1ebd c\u00e1c tuy\u1ebfn DEF v\u1edbi (A) , v\u1ebd c\u00e1t tuy\u1ebfn DM N v\u1edbi (B) . Ch\u1ee9ng minh r\u1eb1ng: 1) DM.DN = DC2; 2) \u25b3DEM \u25b3DN F . B\u00e0i 20. Cho tam gi\u00e1c ABC c\u00f3 tr\u1ef1c t\u00e2m H n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n (O; R) , v\u1ebd \u0111\u01b0\u1eddng k\u00ednh BD c\u1ee7a (O) . 1) Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c AHCD l\u00e0 h\u00ecnh b\u00ecnh h\u00e0nh; 2) T\u00ednh BAC n\u1ebfu bi\u1ebft AH = R. \u221a B\u00e0i 21. Cho tam gi\u00e1c ABC c\u00f3 tr\u1ef1c t\u00e2m H n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n (O) bi\u1ebft AH = R 2. V\u1ebd \u0111\u01b0\u1eddng k\u00ednh BD. 1) Ch\u1ee9ng minh r\u1eb1ng AH = CD; 2) T\u00ednh BAC. B\u00e0i 22. Cho tam gi\u00e1c ABC c\u00f3 tr\u1ef1c t\u00e2m H n\u1ed9i ti\u1ebfp (O; R) . Bi\u1ebft A = 60\u25e6. V\u1ebd \u0111\u01b0\u1eddng k\u00ednh BD c\u1ee7a (O) . 1) Ch\u1ee9ng minh r\u1eb1ng AH = CD; 2) T\u00ednh AH theo R. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 88 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 B\u00e0i 3 Ph\u01b0\u01a1ng t\u00edch I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M Ph\u01b0\u01a1ng t\u00edch m\u1ed9t \u0111i\u1ec3m v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n: N\u1ebfu m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng c\u1eaft \u0111\u01b0\u1eddng tr\u00f2n t\u1ea1i hai \u0111i\u1ec3m ph\u00e2n bi\u1ec7t th\u00ec t\u00edch s\u1ed1 kho\u1ea3ng c\u00e1ch t\u1eeb m\u1ed9t \u0111i\u1ec3m b\u1ea5t k\u1ef3 n\u1eb1m tr\u00ean \u0111\u01b0\u1eddng th\u1eb3ng \u0111\u1ebfn c\u00e1c giao \u0111i\u1ec3m l\u00e0 kh\u00f4ng \u0111\u1ed5i. II. B\u00e0i t\u1eadp: B\u00e0i 1. Cho \u0111\u01b0\u1eddng tr\u00f2n (O) c\u00f3 hai d\u00e2y cung AC v\u00e0 BD c\u1eaft nhau t\u1ea1i \u0111i\u1ec3m I b\u00ean trong (O). Ch\u1ee9ng minh r\u1eb1ng IA.IC = IB.ID B\u00e0i 2. Cho (O) c\u00f3 hai d\u00e2y cung AC v\u00e0 BD k\u00e9o d\u00e0i c\u1eaft nhau t\u1ea1i \u0111i\u1ec3m I b\u00ean ngo\u00e0i (O). Ch\u1ee9ng minh r\u1eb1ng IA.IC = IB.ID. B\u00e0i 3. Tr\u00ean ti\u1ebfp tuy\u1ebfn t\u1ea1i A c\u1ee7a (O) l\u1ea5y \u0111i\u1ec3m B, v\u1ebd c\u00e1t tuy\u1ebfn BCD v\u1edbi (O). Ch\u1ee9ng minh r\u1eb1ng BC.BD = BA2. B\u00e0i 4. Cho \u0111\u01b0\u1eddng tr\u00f2n (O) v\u00e0 \u0111i\u1ec3m M n\u1eb1m b\u00ean ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n. Qua M k\u1ebb ti\u1ebfp tuy\u1ebfn M T v\u00e0 c\u00e1t tuy\u1ebfn M AB, bi\u1ebft M A = 6cm; AB = 12cm. T\u00ednh M T. B\u00e0i 5. Qua \u0111i\u1ec3m M n\u1eb1m ngo\u00e0i (O; R) k\u1ebb ti\u1ebfp tuy\u1ebfn M T (T l\u00e0 ti\u1ebfp \u0111i\u1ec3m) v\u00e0 c\u00e1t tuy\u1ebfn M AB (A, B thu\u1ed9c (O; R)). Bi\u1ebft M T = 6cm, M A = 4cm. T\u00ednh b\u00e1n k\u00ednh R. B\u00e0i 6. Cho \u0111\u01b0\u1eddng tr\u00f2n (O), v\u1ebd hai c\u00e1t tuy\u1ebfn M N P v\u00e0 M EF t\u1edbi \u0111\u01b0\u1eddng tr\u00f2n. Ch\u1ee9ng minh r\u1eb1ng M EB = M AF . B\u00e0i 7. Tr\u00ean d\u00e2y cung BC c\u1ee7a (O; R) l\u1ea5y \u0111i\u1ec3m I b\u1ea5t k\u1ef3. Ch\u1ee9ng minh r\u1eb1ng IB.IC = R2 \u2212 OI2. B\u00e0i 8. Cho m\u1ed9t \u0111i\u1ec3m A b\u00ean ngo\u00e0i (O; R), qua A v\u1ebd c\u00e1t tuy\u1ebfn ABC v\u1edbi (O). Ch\u1ee9ng minh r\u1eb1ng AB.AC = AO2 \u2212 R2. B\u00e0i 9. Cho hai \u0111\u01b0\u1eddng tr\u00f2n (O) v\u00e0 (I) c\u1eaft nhau t\u1ea1i A v\u00e0 B. Tr\u00ean \u0111o\u1ea1n AB l\u1ea5y D, qua D v\u1ebd d\u00e2y cung EF c\u1ee7a I, v\u1ebd d\u00e2y cung N M c\u1ee7a (O). Ch\u1ee9ng minh r\u1eb1ng: 1) DA.DB = DM.DN ; 2) DF.DE = DM.DN. B\u00e0i 10. Cho hai \u0111\u01b0\u1eddng tr\u00f2n (O) v\u00e0 (I) c\u1eaft nhau t\u1ea1i A v\u00e0 B. Tr\u00ean tia \u0111\u1ed1i c\u1ee7a tia AB l\u1ea5y D. T\u1eeb D v\u1ebd c\u00e1t tuy\u1ebfn DEF v\u1edbi (I), v\u1ebd c\u00e1t tuy\u1ebfn DM N v\u1edbi (O). Ch\u1ee9ng minh r\u1eb1ng DE.DF = DM.DN . | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 89","| C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT | Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m B\u00e0i 11. Cho hai \u0111\u01b0\u1eddng tr\u00f2n (O) v\u00e0 (I) c\u1eaft nhau t\u1ea1i A v\u00e0 B. G\u1ecdi M N l\u00e0 ti\u1ebfp tuy\u1ebfn chung ngo\u00e0i c\u1ee7a (O) v\u00e0 (I), M thu\u1ed9c (O), N thu\u1ed9c (I), D l\u00e0 giao \u0111i\u1ec3m c\u1ee7a M N v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng AB. Ch\u1ee9ng minh r\u1eb1ng D l\u00e0 trung \u0111i\u1ec3m c\u1ee7a M N. B\u00e0i 12. Cho hai \u0111\u01b0\u1eddng tr\u00f2n (O) v\u00e0 (I) c\u1eaft nhau t\u1ea1i A v\u00e0 B. Tr\u00ean tia \u0111\u1ed1i tia AB l\u1ea5y \u0111i\u1ec3m D, t\u1eeb D v\u1ebd c\u00e1c ti\u1ebfp tuy\u1ebfn DM v\u1edbi (O), ti\u1ebfp tuy\u1ebfn DN v\u1edbi (I). Ch\u1ee9ng minh r\u1eb1ng DM = DN . B\u00e0i 13. Cho n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n (O) \u0111\u01b0\u1eddng k\u00ednh AB, d\u00e2y cung AB song song v\u1edbi CD (D thu\u1ed9c cung AC). Qua A k\u1ebb ti\u1ebfp tuy\u1ebfn v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n c\u1eaft CD t\u1ea1i H. Ch\u1ee9ng minh r\u1eb1ng AH2 = HC.HD. B\u00e0i 14. Cho (O) c\u00f3 \u0111\u01b0\u1eddng k\u00ednh AB. G\u1ecdi C l\u00e0 m\u1ed9t \u0111i\u1ec3m tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n, ti\u1ebfp tuy\u1ebfn t\u1ea1i A v\u00e0 C c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n c\u1eaft nhau \u1edf M, c\u00e1c tia BC v\u00e0 AM c\u1eaft nhau \u1edf N. Ch\u1ee9ng minh r\u1eb1ng M N = M C. B\u00e0i 15. Cho tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp (O). G\u1ecdi D l\u00e0 m\u1ed9t \u0111i\u1ec3m tr\u00ean c\u1ea1nh BC, tia AD c\u1eaft cung BC \u1edf E. Ch\u1ee9ng minh r\u1eb1ng AD.CE = AB.AD. B\u00e0i 16. Cho \u0111\u01b0\u1eddng tr\u00f2n (O) c\u00f3 d\u00e2y BC song song v\u1edbi ti\u1ebfp tuy\u1ebfn t\u1ea1i A c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n. L\u1ea5y \u0111i\u1ec3m E thu\u1ed9c cung BC kh\u00f4ng ch\u1ee9a A. Tia CE c\u1eaft ti\u1ebfp tuy\u1ebfn t\u1ea1i M, M B c\u1eaft (O) t\u1ea1i D. Tia ED c\u1eaft \u0111o\u1ea1n th\u1eb3ng AM t\u1ea1i I. Ch\u1ee9ng minh r\u1eb1ng IM 2 = IE.ID. B\u00e0i 17. Cho \u0111i\u1ec3m M n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O. V\u1ebd hai ti\u1ebfp tuy\u1ebfn M A v\u00e0 M C c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n (O) (A v\u00e0 C l\u00e0 hai ti\u1ebfp \u0111i\u1ec3m). V\u1ebd d\u00e2y cung BC song song v\u1edbi AM, M B c\u1eaft (O) t\u1ea1i D, CD c\u1eaft AM t\u1ea1i I. Ch\u1ee9ng minh r\u1eb1ng IM 2 = IC.ID. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 4 T\u1ee9 gi\u00e1c n\u1ed9i ti\u1ebfp I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: 1. \u0110\u1ecbnh ngh\u0129a: T\u1ee9 gi\u00e1c n\u1ed9i ti\u1ebfp l\u00e0 t\u1ee9 gi\u00e1c c\u00f3 4 \u0111\u1ec9nh n\u1eb1m tr\u00ean m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n. 2. T\u00ednh ch\u1ea5t: T\u1ee9 gi\u00e1c n\u1ed9i ti\u1ebfp c\u00f3: Hai g\u00f3c \u0111\u1ed1i b\u00f9 nhau. G\u00f3c ngo\u00e0i t\u1ea1i m\u1ed9t \u0111\u1ec9nh b\u1eb1ng g\u00f3c \u0111\u1ed1i trong. Hai \u0111\u1ec9nh li\u00ean ti\u1ebfp c\u00f9ng nh\u00ecn m\u1ed9t c\u1ea1nh d\u01b0\u1edbi hai g\u00f3c b\u1eb1ng nhau th\u00ec b\u1eb1ng nhau. 3. C\u00e1ch ch\u1ee9ng minh t\u1ee9 gi\u00e1c n\u1ed9i ti\u1ebfp: Ch\u1ee9ng minh t\u1ee9 gi\u00e1c c\u00f3 b\u1ed1n \u0111\u1ec9nh c\u00e1ch \u0111\u1ec1u m\u1ed9t \u0111i\u1ec3m. Ch\u1ee9ng minh t\u1ee9 gi\u00e1c c\u00f3 hai g\u00f3c \u0111\u1ed1i b\u00f9 nhau. 90 N\u0103m h\u1ecdc: 2023 - 2024","S GI\u00c1O TR\u00ccNH TO\u00c1N 9 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M S Ch\u1ee9ng minh t\u1ee9 gi\u00e1c c\u00f3 g\u00f3c ngo\u00e0i b\u1eb1ng g\u00f3c \u0111\u1ed1i trong. Ch\u1ee9ng minh t\u1ee9 gi\u00e1c c\u00f3 hai \u0111\u1ec9nh li\u00ean ti\u1ebfp c\u00f9ng nh\u00ecn m\u1ed9t c\u1ea1nh d\u01b0\u1edbi hai g\u00f3c b\u1eb1ng nhau. 4. M\u1ed9t s\u1ed1 \u0111\u1ecbnh l\u00fd n\u00e2ng cao: \u0110\u1ecbnh l\u00fd Ptoleme: Trong m\u1ed9t t\u1ee9 gi\u00e1c n\u1ed9i ti\u1ebfp t\u1ed5ng c\u00e1c t\u00edch c\u1ee7a hai c\u1ea1nh \u0111\u1ed1i di\u1ec7n b\u1eb1ng t\u00edch hai \u0111\u01b0\u1eddng ch\u00e9o. \u0110\u01b0\u1eddng tr\u00f2n Euler: Trong m\u1ed9t tam gi\u00e1c, c\u00e1c ch\u00e2n \u0111\u01b0\u1eddng cao, trung \u0111i\u1ec3m c\u00e1c c\u1ea1nh, trung \u0111i\u1ec3m c\u00e1c \u0111o\u1ea1n n\u1ed1i tr\u1ef1c t\u00e2m \u0111\u1ebfn \u0111\u1ec9nh c\u00f9ng n\u1eb1m tr\u00ean m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n. \u0110\u01b0\u1eddng th\u1eb3ng Sim-son: H\u00ecnh chi\u1ebfu c\u1ee7a m\u1ed9t \u0111i\u1ec3m b\u1ea5t k\u1ef3 thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n l\u00ean ba c\u1ea1nh c\u1ee7a m\u1ed9t tam gi\u00e1c n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n c\u00f9ng n\u1eb1m tr\u00ean m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng. II. B\u00e0i t\u1eadp: B\u00e0i 1. Cho t\u1ee9 gi\u00e1c ABCD n\u1ed9i ti\u1ebfp, bi\u1ebft A = 80\u25e6, B = 100\u25e6. T\u00ednh s\u1ed1 \u0111o C, D. B\u00e0i 2. Cho t\u1ee9 gi\u00e1c ABCD n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n (O) c\u00f3 hai \u0111\u01b0\u1eddng ch\u00e9o AC v\u00e0 BD c\u1eaft nhau t\u1ea1i E. Ch\u1ee9ng minh r\u1eb1ng \u25b3AEB \u25b3CED. B\u00e0i 3. Cho t\u1ee9 gi\u00e1c ABCD n\u1ed9i ti\u1ebfp c\u00f3 hai c\u1ea1nh AB v\u00e0 CD k\u00e9o d\u00e0i c\u1eaft nhau t\u1ea1i E. Ch\u1ee9ng minh r\u1eb1ng \u25b3EAC \u25b3EBD. B\u00e0i 4. Cho t\u1ee9 gi\u00e1c ABCD c\u00f3 A = C = 90\u25e6. G\u1ecdi I l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AC v\u00e0 BD. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c ABCD n\u1ed9i ti\u1ebfp. B\u00e0i 5. Cho tam gi\u00e1c ABC nh\u1ecdn, c\u00e1c \u0111\u01b0\u1eddng cao BE, CF c\u1eaft nhau t\u1ea1i H v\u00e0 c\u1eaft \u0111\u01b0\u1eddng tr\u00f2n l\u1ea7n l\u01b0\u1ee3t t\u1ea1i M, N, P. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c BCEF v\u00e0 AF HE n\u1ed9i ti\u1ebfp. B\u00e0i 6. Cho tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n (O) c\u00f3 AB l\u00e0 \u0111\u01b0\u1eddng k\u00ednh. T\u1eeb \u0111i\u1ec3m H thu\u1ed9c c\u1ea1nh AC, k\u1ebb HK\u22a5AB t\u1ea1i K. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c CHKB n\u1ed9i ti\u1ebfp. B\u00e0i 7. Cho tam gi\u00e1c ABC nh\u1ecdn c\u00f3 ba \u0111\u01b0\u1eddng cao AD, BE, CF c\u1eaft nhau t\u1ea1i H. Ch\u1ee9ng minh r\u1eb1ng: 1) C\u00e1c t\u1ee9 gi\u00e1c AEHF, CEHD n\u1ed9i ti\u1ebfp; 2) C\u00e1c t\u1ee9 gi\u00e1c BF EC, AEDB n\u1ed9i ti\u1ebfp. B\u00e0i 8. Cho t\u1ee9 gi\u00e1c ABCD c\u00f3 A = 110\u25e6, C = 70\u25e6. G\u1ecdi I l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AC v\u00e0 BD. Ch\u1ee9ng minh r\u1eb1ng: 1) T\u1ee9 gi\u00e1c ABCD n\u1ed9i ti\u1ebfp; 2) IA.IC = IB.ID. B\u00e0i 9. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH; g\u1ecdi I, K l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AC v\u00e0 HC. Ch\u1ee9ng minh r\u1eb1ng: 1) IK\/\/AH; | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 91","| C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT | Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m 2) T\u1ee9 gi\u00e1c BAIK n\u1ed9i ti\u1ebfp. B\u00e0i 10. Cho tam gi\u00e1c ABC l\u1ea5y D thu\u1ed9c AB v\u00e0 E thu\u1ed9c AC sao cho AE.AC = AB.AD. 1) Ch\u1ee9ng minh r\u1eb1ng \u25b3ADE \u0111\u1ed3ng d\u1ea1ng v\u1edbi \u25b3ACB; 2) Ch\u1ee9ng minh r\u1eb1ng T\u1ee9 gi\u00e1c BDEC n\u1ed9i ti\u1ebfp; 3) Hai \u0111\u01b0\u1eddng th\u1eb3ng DE v\u00e0 BC c\u1eaft nhau t\u1ea1i I. Ch\u1ee9ng minh r\u1eb1ng IB.IC = ID.IE. B\u00e0i 11. Cho \u25b3ABC nh\u1ecdn c\u00f3 \u0111\u01b0\u1eddng cao AD, g\u1ecdi E v\u00e0 F l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a D l\u00ean hai c\u1ea1nh AB v\u00e0 AC. 1) Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c AEDF n\u1ed9i ti\u1ebfp; 2) Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c BEF C n\u1ed9i ti\u1ebfp; 3) G\u1ecdi I l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AD v\u00e0 EF. Ch\u1ee9ng minh r\u1eb1ng IA.ID = IE.IF. B\u00e0i 12. Cho tam gi\u00e1c ABC nh\u1ecdn n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n O. V\u1ebd \u0111\u01b0\u1eddng th\u1eb3ng d song song v\u1edbi ti\u1ebfp tuy\u1ebfn Ax c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n v\u00e0 c\u1eaft hai d\u00e2y AB, AC l\u1ea7n l\u01b0\u1ee3t t\u1ea1i M v\u00e0 N (M kh\u00f4ng tr\u00f9ng v\u1edbi B, N kh\u00f4ng tr\u00f9ng v\u1edbi C). Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c BM N C n\u1ed9i ti\u1ebfp. B\u00e0i 13. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A, l\u1ea5y \u0111i\u1ec3m M tr\u00ean c\u1ea1nh BC, tia ph\u00e2n gi\u00e1c c\u1ee7a AM B c\u1eaft c\u1ea1nh AB t\u1ea1i E, tia ph\u00e2n gi\u00e1c c\u1ee7a AM C c\u1eaft c\u1ea1nh AC t\u1ea1i F. 1) Ch\u1ee9ng minh r\u1eb1ng T\u1ee9 gi\u00e1c AEM F n\u1ed9i ti\u1ebfp; 2) G\u1ecdi I l\u00e0 giao \u0111i\u1ec3m c\u1ee7a EF v\u00e0 AM. Ch\u1ee9ng minh r\u1eb1ng IA.IM = IF.IE. B\u00e0i 14. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A c\u00f3 AI l\u00e0 \u0111\u01b0\u1eddng trung tuy\u1ebfn. Tr\u00ean AB, AC l\u1ea5y E v\u00e0 F sao cho EF vu\u00f4ng g\u00f3c v\u1edbi AI. Ch\u1ee9ng minh r\u1eb1ng: 1) Tam gi\u00e1c AIC c\u00e2n t\u1ea1i I; 2) T\u1ee9 gi\u00e1c BEF C n\u1ed9i ti\u1ebfp. B\u00e0i 15. Cho tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp (O), g\u1ecdi xy l\u00e0 ti\u1ebfp tuy\u1ebfn qua A c\u1ee7a (O). L\u1ea5y D, E thu\u1ed9c c\u1ea1nh AC, AB sao cho DE\/\/xy. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c BEDC n\u1ed9i ti\u1ebfp. B\u00e0i 16. Cho tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp (O), g\u1ecdi D l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a B l\u00ean AC, E l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a C l\u00ean AB. Ch\u1ee9ng minh r\u1eb1ng: 1) T\u1ee9 gi\u00e1c BEDC n\u1ed9i ti\u1ebfp; 2) ED song song v\u1edbi ti\u1ebfp tuy\u1ebfn t\u1ea1i A. B\u00e0i 17. Cho tam gi\u00e1c ABC nh\u1ecdn n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n (O). Tr\u00ean AB l\u1ea5y D, tr\u00ean AC l\u1ea5y E sao cho DE vu\u00f4ng g\u00f3c v\u1edbi AO. T\u1eeb A v\u1ebd ti\u1ebfp tuy\u1ebfn xAy v\u1edbi (O). Ch\u1ee9ng minh r\u1eb1ng: 1) xy\/\/DE; 2) T\u1ee9 gi\u00e1c BDEC n\u1ed9i ti\u1ebfp. B\u00e0i 18. Cho tam gi\u00e1c ABC nh\u1ecdn n\u1ed9i ti\u1ebfp (O) c\u00f3 AB < AC. AO c\u1eaft BC t\u1ea1i I. \u0110\u01b0\u1eddng th\u1eb3ng qua 92 N\u0103m h\u1ecdc: 2023 - 2024","S GI\u00c1O TR\u00ccNH TO\u00c1N 9 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M I vu\u00f4ng g\u00f3c v\u1edbi AO c\u1eaft AC t\u1ea1i E, c\u1eaft tia AB t\u1ea1i D. V\u1ebd \u0111\u01b0\u1eddng k\u00ednh AF c\u1ee7a (O). Ch\u1ee9ng minh r\u1eb1ng: 1) T\u1ee9 gi\u00e1c BIF D n\u1ed9i ti\u1ebfp; 2) T\u1ee9 gi\u00e1c DBEC n\u1ed9i ti\u1ebfp. B\u00e0i 19. Cho t\u1ee9 gi\u00e1c ABCD n\u1ed9i ti\u1ebfp (O). G\u1ecdi M, N l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a D l\u00ean hai \u0111\u01b0\u1eddng th\u1eb3ng ch\u1ee9a c\u1ea1nh AB v\u00e0 AC. Ch\u1ee9ng minh r\u1eb1ng: 1) T\u1ee9 gi\u00e1c AM DN n\u1ed9i ti\u1ebfp; 2) \u25b3DBC \u25b3DM N . B\u00e0i 20. Cho t\u1ee9 gi\u00e1c ABCD n\u1ed9i ti\u1ebfp (O). G\u1ecdi M, N l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a D l\u00ean hai \u0111\u01b0\u1eddng th\u1eb3ng ch\u1ee9a c\u1ea1nh AB v\u00e0 AC. G\u1ecdi I l\u00e0 giao \u0111i\u1ec3m M N v\u00e0 BC. Ch\u1ee9ng minh r\u1eb1ng DI\u22a5BC. B\u00e0i 21. Cho t\u1ee9 gi\u00e1c ABCD n\u1ed9i ti\u1ebfp (O). G\u1ecdi M, N, I l\u1ea7n l\u01b0\u1ee3t l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a D l\u00ean c\u00e1c \u0111\u01b0\u1eddng th\u1eb3ng ch\u1ee9a c\u1ea1nh AB, AC v\u00e0 BC. Ch\u1ee9ng minh r\u1eb1ng: 1) C\u00e1c t\u1ee9 gi\u00e1c: N CID, BIDM n\u1ed9i ti\u1ebfp; 2) \u25b3BM D v\u00e0 \u25b3CN D \u0111\u1ed3ng d\u1ea1ng; 3) M, N, I th\u1eb3ng h\u00e0ng. B\u00e0i 22. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH, tia ph\u00e2n gi\u00e1c BE c\u1ee7a ABC c\u1eaft AH t\u1ea1i M v\u00e0 c\u1eaft tia ph\u00e2n gi\u00e1c AI c\u1ee7a HAC t\u1ea1i N. Ch\u1ee9ng minh r\u1eb1ng: 1) T\u1ee9 gi\u00e1c M N IH n\u1ed9i ti\u1ebfp; 2) T\u1ee9 gi\u00e1c AN HB n\u1ed9i ti\u1ebfp; 3) T\u1ee9 gi\u00e1c CEN H n\u1ed9i ti\u1ebfp; 4) T\u1ee9 gi\u00e1c AEIB n\u1ed9i ti\u1ebfp. B\u00e0i 23. Cho h\u00ecnh vu\u00f4ng ABCD g\u1ecdi E, F l\u1ea7n l\u01b0\u1ee3c l\u00e0 trung \u0111i\u1ec3m c\u1ee7a DC v\u00e0 DA, BE v\u00e0 CF c\u1eaft nhau t\u1ea1i I. Ch\u1ee9ng minh r\u1eb1ng: 1) \u25b3BCE = \u25b3CDF ; 2) T\u1ee9 gi\u00e1c ABIF n\u1ed9i ti\u1ebfp. B\u00e0i 24. Cho b\u1ed1n \u0111i\u1ec3m A, B, C, D theo th\u1ee9 t\u1ef1 \u0111\u00f3 c\u00f9ng thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n (O) sao cho AB k\u00e9o d\u00e0i c\u1eaft DC k\u00e9o d\u00e0i t\u1ea1i E. G\u1ecdi F l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BE, G l\u00e0 trung \u0111i\u1ec3m c\u1ee7a CE. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c AF GD n\u1ed9i ti\u1ebfp. B\u00e0i 25. Cho b\u1ed1n \u0111i\u1ec3m A, B, C, D theo th\u1ee9 t\u1ef1 \u0111\u00f3 c\u00f9ng thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n (O). G\u1ecdi M l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AC v\u00e0 BD, g\u1ecdi K l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AM, L l\u00e0 trung \u0111i\u1ec3m c\u1ee7a DM. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c BKLC n\u1ed9i ti\u1ebfp. B\u00e0i 26. Cho b\u1ed1n \u0111i\u1ec3m A, B, C, D theo th\u1ee9 t\u1ef1 \u0111\u00f3 c\u00f9ng thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n (O). G\u1ecdi M l\u00e0 giao | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 93","| C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT | Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m \u0111i\u1ec3m c\u1ee7a AC v\u00e0 BD, g\u1ecdi K, L, E, F l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a M A, M D, M C v\u00e0 M B. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c KLEF n\u1ed9i ti\u1ebfp. B\u00e0i 27. Cho b\u1ed1n \u0111i\u1ec3m A, B, C, D theo th\u1ee9 t\u1ef1 \u0111\u00f3 c\u00f9ng thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n (O). G\u1ecdi I l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AB v\u00e0 CD; g\u1ecdi E, F, G, H l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a IA, ID, IB v\u00e0 IC. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c EGHF n\u1ed9i ti\u1ebfp. B\u00e0i 28. Cho (O), t\u1eeb C n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n, v\u1ebd hai ti\u1ebfp tuy\u1ebfn CD v\u00e0 CE, c\u00e1t tuy\u1ebfn CBA. G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m AB. Ch\u1ee9ng minh r\u1eb1ng: 1) 4 \u0111i\u1ec3m O, I, D, C c\u00f9ng thu\u1ed9c m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n; 2) 4 \u0111i\u1ec3m O, I, D, E c\u00f9ng thu\u1ed9c m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n. B\u00e0i 29. Cho (O) c\u00f3 hai ti\u1ebfp tuy\u1ebfn t\u1ea1i D v\u00e0 E c\u1eaft nhau t\u1ea1i C. V\u1ebd c\u00e1t tuy\u1ebfn CBA. G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m AB. 1) Ch\u1ee9ng minh r\u1eb1ng 5 \u0111i\u1ec3m O, I, D, C, E c\u00f9ng thu\u1ed9c m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n; 2) Ch\u1ee9ng minh r\u1eb1ng IC l\u00e0 tia ph\u00e2n gi\u00e1c c\u1ee7a DIE. B\u00e0i 30. Cho (O) c\u00f3 d\u00e2y CD vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng k\u00ednh AB t\u1ea1i H, l\u1ea5y E thu\u1ed9c cung nh\u1ecf AC, EB c\u1eaft AC t\u1ea1i F, ED c\u1eaft AB t\u1ea1i G. Ch\u1ee9ng minh r\u1eb1ng: 1) T\u1ee9 gi\u00e1c EF GA n\u1ed9i ti\u1ebfp; 2) Ba \u0111\u01b0\u1eddng th\u1eb3ng EA, F G, CB \u0111\u1ed3ng quy. B\u00e0i 31. Cho n\u1eeda (O) \u0111\u01b0\u1eddng k\u00ednh AB. I thu\u1ed9c \u0111o\u1ea1n AO. L\u1ea5y E thu\u1ed9c n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n (O). \u0110\u01b0\u1eddng th\u1eb3ng qua E vu\u00f4ng g\u00f3c v\u1edbi IE c\u1eaft hai ti\u1ebfp tuy\u1ebfn t\u1ea1i A v\u00e0 B (c\u00f9ng ph\u00eda v\u1edbi n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n) l\u1ea7n l\u01b0\u1ee3t t\u1ea1i hai \u0111i\u1ec3m C v\u00e0 D. Ch\u1ee9ng minh r\u1eb1ng \u25b3CID vu\u00f4ng t\u1ea1i I. B\u00e0i 32. Cho (O) c\u00f3 hai ti\u1ebfp tuy\u1ebfn t\u1ea1i B v\u00e0 C c\u1eaft nhau t\u1ea1i A. L\u1ea5y M tr\u00ean d\u00e2y cung BC c\u1ee7a (O) sao cho M B < M C. \u0110\u01b0\u1eddng th\u1eb3ng qua M v\u00e0 vu\u00f4ng g\u00f3c OM c\u1eaft ti\u1ebfp tuy\u1ebfn AB k\u00e9o d\u00e0i t\u1ea1i I, c\u1eaft ti\u1ebfp tuy\u1ebfn AC t\u1ea1i K. Ch\u1ee9ng minh r\u1eb1ng M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a KI. B\u00e0i 33. Cho h\u00ecnh thang vu\u00f4ng ABCD c\u00f3 AB\/\/DC v\u00e0 DC = 2AB, g\u1ecdi M l\u00e0 trung \u0111i\u1ec3m DC, E l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a M l\u00ean c\u1ea1nh AC. Ch\u1ee9ng minh r\u1eb1ng: 1) T\u1ee9 gi\u00e1c AEM D v\u00e0 ABM D n\u1ed9i ti\u1ebfp; 2) DE vu\u00f4ng g\u00f3c BE. B\u00e0i 34. Cho (O) c\u00f3 ti\u1ebfp tuy\u1ebfn t\u1ea1i B v\u00e0 C c\u1eaft nhau t\u1ea1i A, OA c\u1eaft (O) t\u1ea1i D. Tr\u00ean cung l\u1edbn BC l\u1ea5y hai \u0111i\u1ec3m G, H, GD c\u1eaft BC t\u1ea1i E, HD c\u1eaft BC t\u1ea1i F. Ch\u1ee9ng minh r\u1eb1ng: 1) D l\u00e0 \u0111i\u1ec3m ch\u00ednh gi\u1eefa c\u1ee7a cung nh\u1ecf BC; 2) T\u1ee9 gi\u00e1c GEF H n\u1ed9i ti\u1ebfp. B\u00e0i 35. T\u1ee9 gi\u00e1c ABCD c\u00f3 hai \u0111\u01b0\u1eddng ch\u00e9o AC v\u00e0 BD c\u1eaft nhau t\u1ea1i O, c\u00f3 ABD = ACD. G\u1ecdi E l\u00e0 giao \u0111i\u1ec3m c\u1ee7a hai \u0111\u01b0\u1eddng th\u1eb3ng AD v\u00e0 BC. Ch\u1ee9ng minh r\u1eb1ng EA.EB = ED.EC. 94 N\u0103m h\u1ecdc: 2023 - 2024"]