Hướng dẫn giải Bài 97 (Trang 105 SGK Toán 9 Hình học, Tập 2)
<p>Cho tam giác ABC vuông ở A. Trên AC lấy một điểm M và vẽ đường tròn đường kính MC. Kẻ Bm cắt đường tròn tại D. Đường thẳng DA cắt đường tròn tại S. Chứng minh rằng :</p>
<p>a) ABCD là một tứ giác nội tiếp ;<span class="mce-nbsp-wrap" contenteditable="false"> </span><span class="mce-nbsp-wrap" contenteditable="false"> </span><span class="mce-nbsp-wrap" contenteditable="false"> </span><span class="mce-nbsp-wrap" contenteditable="false"> </span><span class="mce-nbsp-wrap" contenteditable="false"> </span><span class="mce-nbsp-wrap" contenteditable="false"> </span><span class="mce-nbsp-wrap" contenteditable="false"> </span>b)<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>B</mi><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><mover><mrow><mi>A</mi><mi>C</mi><mi>D</mi></mrow><mo>^</mo></mover></math>;</p>
<p>c) CA là tia phân giác của góc SCB.</p>
<p><strong>Giải :</strong></p>
<p>a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>M</mi><mi>O</mi><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><mn>90</mn><mo>°</mo></math>(góc nội tiếp chắn nửa đường tròn)</p>
<p><span class="mce-nbsp-wrap" contenteditable="false"> </span><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>B</mi><mi>A</mi><mi>C</mi></mrow><mo>^</mo></mover><mo>=</mo><mn>90</mn><mo>°</mo></math> (theo giả thiết)<img class="wscnph" 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width="191" height="156" /></p>
<p>Điểm A và D đều nhìn đoạn thẳng BC cố định dưới góc <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo>°</mo></math>, vậy A và D cùng nằm trên đường tròn đường kính BC.</p>
<p>Nói cách khác, ABCD là một tứ giác nội tiếp đường tròn đường kính BC.</p>
<p>b) Trong đường tròn đường kính BC, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>B</mi><mi>D</mi></mrow><mo>^</mo></mover><mo>=</mo><mover><mrow><mi>A</mi><mi>C</mi><mi>D</mi></mrow><mo>^</mo></mover></math> vì cùng chắn cung <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>⏜</mo></mover></math>.</p>
<p>c) <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>S</mi><mi>D</mi><mi>M</mi></mrow><mo>^</mo></mover><mo>=</mo><mover><mrow><mi>M</mi><mi>C</mi><mi>S</mi></mrow><mo>^</mo></mover><mo> </mo><mo>(</mo><mn>1</mn><mo>)</mo></math> (cùng chắn cung <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>M</mi><mi>S</mi></mrow><mo>⏜</mo></mover></math> của đường tròn (O). Lại có </p>
<p><span class="mce-nbsp-wrap" contenteditable="false"> </span> <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>D</mi><mi>B</mi></mrow><mo>^</mo></mover><mo>=</mo><mover><mrow><mi>A</mi><mi>C</mi><mi>B</mi></mrow><mo>^</mo></mover><mo> </mo><mo>(</mo><mn>2</mn><mo>)</mo></math> (cùng chắn cung <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>⏜</mo></mover></math> của đường tròn đường kính BC).</p>
<p>So sánh (1) và (2) suy ra:</p>
<p><span class="mce-nbsp-wrap" contenteditable="false"> </span><span class="mce-nbsp-wrap" contenteditable="false"> </span><span class="mce-nbsp-wrap" contenteditable="false"> </span><span class="mce-nbsp-wrap" contenteditable="false"> </span><span class="mce-nbsp-wrap" contenteditable="false"> </span><span class="mce-nbsp-wrap" contenteditable="false"> </span><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>S</mi><mi>C</mi><mi>A</mi></mrow><mo>^</mo></mover><mo>=</mo><mover><mrow><mi>A</mi><mi>C</mi><mi>B</mi></mrow><mo>^</mo></mover></math></p>
<p>Vậy CA là tia phân giác của <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>S</mi><mi>C</mi><mi>B</mi></mrow><mo>^</mo></mover><mo>.</mo></math></p>
<p> </p>
<p><span class="mce-nbsp-wrap" contenteditable="false"> </span></p>
Hướng dẫn Giải Bài 97 (Trang 105, SGK Toán Hình học 9, Tập 2)