lines form equal angles with line, 90^o (2019 YuMSh Olympiad Finals 8.6)

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High School Math Grades 9-12, Ages 13-18

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lines form equal angles with line, 90^o (2019 YuMSh Olympiad Finals 8.6)

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#1 h Mar 4, 2021, 8:16 PM

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In a right-angled triangle $ABC$ , point $M$ is the midpoint of the hypotenuse $AB$ . On the ray $CM$ mark the points $X$ and $Y$ . It is known that lines $XA$ and $YA$ form the same angles with line $AC$ . Prove that lines $XB$ and $YB$ form the same angles with line $BC$ .

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#2 h Mar 5, 2021, 7:59 PM

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Given $A(0,a),B(b,0),C(0,0)$ and $M(\frac{b}{2},\frac{a}{2})$ .

Point $X(\lambda,\frac{a\lambda}{b}) \in CM$ .
Condition $Y(\frac{b\lambda}{2\lambda-b},\frac{a\lambda}{2\lambda-b})$ .

Slope of the line $XB\ :\ m_{XB}=\frac{a\lambda}{b(\lambda-b)}$ , slope of the line $YB\ :\ m_{YB}=\frac{a\lambda}{b(b-\lambda)}$ .

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