Hướng dẫn giải Bài 23 (Trang 76 SGK Toán 9 Hình học, Tập 2)

Cho đường tròn (O) và một điểm M cố định không nằm trên đường tròn. Qua M kẻ hai đường thẳng . Đường thẳng thứ nhất cắt (O) tại A và B. Đường thẳng thứ hai cắt (O) tại C và D. Chứng minh MA.MB = MC.MD. Hướng dẫn: Xét cả hai trường hợp điểm M nằm bên trong và bên ngoài đường tròn. Trong mỗi trường hợp, xét hai tam giác đồng dạng. Giải: <img class="wscnph" style="max-width: 100%;" src="https://static.colearn.vn:8413/v1.0/upload/library/03052022/6dcc5140-3db9-4a67-a728-eafda1c7cca4.PNG" /> TH1: M nằm trong đường tròn. <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow><mo>^</mo></mover><mo>,</mo><mo> </mo><mover><mrow><mi>B</mi><mi>A</mi><mi>D</mi></mrow><mo>^</mo></mover></math> là hai góc nội tiếp cùng chắn cung <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>^</mo></mover></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mover><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><mo> </mo><mover><mrow><mi>B</mi><mi>A</mi><mi>D</mi></mrow><mo>^</mo></mover></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>△</mo><mi>B</mi><mi>M</mi><mi>C</mi><mo> </mo><mi>v</mi><mi>à</mi><mo> </mo><mo>△</mo><mi>D</mi><mi>M</mi><mi>A</mi><mo> </mo><mi>c</mi><mi>ó</mi><mo>:</mo><mspace linebreak="newline"/><mover><mrow><mi>B</mi><mi>C</mi><mi>M</mi></mrow><mo>^</mo></mover><mo>=</mo><mo> </mo><mover><mrow><mi>D</mi><mi>A</mi><mi>M</mi></mrow><mo>^</mo></mover><mo> </mo><mfenced><mrow><mi>c</mi><mi>m</mi><mi>t</mi></mrow></mfenced><mo>;</mo><mspace linebreak="newline"/><mover><mrow><mi>B</mi><mi>M</mi><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><mo> </mo><mover><mrow><mi>D</mi><mi>M</mi><mi>A</mi></mrow><mo>^</mo></mover><mo> </mo><mfenced><mrow><mi>đ</mi><mi>ố</mi><mi>i</mi><mo> </mo><mi>đ</mi><mi>ỉ</mi><mi>n</mi><mi>h</mi></mrow></mfenced><mo>;</mo></math> <img class="wscnph" style="max-width: 100%;" src="data:image/png;base64,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" width="165" height="20" /> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mi>B</mi><mi>M</mi></mrow><mrow><mi>D</mi><mi>M</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>M</mi><mi>C</mi></mrow><mrow><mi>M</mi><mi>A</mi></mrow></mfrac></math> ⇒ MA.MB = MC.MD TH2: M nằm ngoài đường tròn. <img class="wscnph" style="max-width: 100%;" src="https://static.colearn.vn:8413/v1.0/upload/library/03052022/cfa6c9cd-7eb5-4c5c-9a35-cbc72c54675b.PNG" /> <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>B</mi><mi>A</mi><mi>D</mi></mrow><mo>^</mo></mover></math> là góc nội tiếp chắn cung lớn <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>B</mi><mi>m</mi><mi>D</mi></mrow><mo>^</mo></mover></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mover><mrow><mi>B</mi><mi>A</mi><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>.</mo><mi>s</mi><mi>đ</mi><mover><mrow><mi>B</mi><mi>m</mi><mi>D</mi></mrow><mo>^</mo></mover><mo>.</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow><mo>^</mo></mover></math> là góc nội tiếp chắn cung nhỏ <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>B</mi><mi>n</mi><mi>D</mi></mrow><mo>^</mo></mover></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mover><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>.</mo><mi>s</mi><mi>đ</mi><mover><mrow><mi>B</mi><mi>n</mi><mi>D</mi></mrow><mo>^</mo></mover><mo>.</mo><mspace linebreak="newline"/><mo>⇒</mo><mover><mrow><mi>B</mi><mi>A</mi><mi>D</mi></mrow><mo>^</mo></mover><mo> </mo><mo>+</mo><mover><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow><mo>^</mo></mover><mspace linebreak="newline"/><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mi>s</mi><mi>đ</mi><mo> </mo><mover><mrow><mi>B</mi><mi>m</mi><mi>D</mi></mrow><mo>^</mo></mover><mo>+</mo><mi>s</mi><mi>đ</mi><mo> </mo><mover><mrow><mi>B</mi><mi>n</mi><mi>D</mi></mrow><mo>^</mo></mover></mrow></mfenced><mo>=</mo><msup><mn>180</mn><mi>o</mi></msup><mo>.</mo><mspace linebreak="newline"/><mi>M</mi><mi>à</mi><mo> </mo><mover><mrow><mi>B</mi><mi>C</mi><mi>M</mi></mrow><mo>^</mo></mover><mo>+</mo><mover><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><msup><mn>180</mn><mi>o</mi></msup><mfenced><mrow><mi>h</mi><mi>a</mi><mi>i</mi><mo> </mo><mi>g</mi><mi>ó</mi><mi>c</mi><mo> </mo><mi>k</mi><mi>ề</mi><mo> </mo><mi>b</mi><mi>ù</mi></mrow></mfenced><mspace linebreak="newline"/><mo>⇒</mo><mover><mrow><mi>B</mi><mi>A</mi><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><mover><mrow><mi>B</mi><mi>C</mi><mi>M</mi></mrow><mo>^</mo></mover></math> ΔMBC và ΔMDA có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>M</mi><mo> </mo></mrow><mo>^</mo></mover><mi>c</mi><mi>h</mi><mi>u</mi><mi>n</mi><mi>g</mi><mo>,</mo><mo> </mo><mover><mrow><mi>M</mi><mi>C</mi><mi>B</mi></mrow><mo>^</mo></mover><mo>=</mo><mover><mrow><mi>M</mi><mi>A</mi><mi>D</mi></mrow><mo>^</mo></mover></math> <img class="wscnph" style="max-width: 100%;" src="https://static.colearn.vn:8413/v1.0/upload/library/10032022/ca671b9f-d178-4087-99a2-0176562ce389.PNG" /> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mi>M</mi><mi>B</mi></mrow><mrow><mi>M</mi><mi>D</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>M</mi><mi>C</mi></mrow><mrow><mi>M</mi><mi>A</mi></mrow></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mi>M</mi><mi>A</mi><mo>.</mo><mi>M</mi><mi>B</mi><mo>=</mo><mi>M</mi><mi>C</mi><mo>.</mo><mi>M</mi><mi>D</mi><mspace linebreak="newline"/></math>