2 squares on sides of triangle - 2021 Bulgaria JBMO TST 1.2

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

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2 squares on sides of triangle - 2021 Bulgaria JBMO TST 1.2

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#1 h Nov 3, 2022, 9:29 PM

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On the sides $BC$ and $CA$ of the acute $\vartriangle ABC$ , outside of the triangle, the squares $CBA_1M$ and $ACNB_1$ are constructed. Prove that the intersection point of $AA_1$ and $BB_1$ lies on the height $CC_1$ of $\vartriangle ABC$ .

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#2 h Nov 4, 2022, 5:24 PM

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Given $\triangle ABC\ :\ A(a,b),B(0,0),C(c,0)$ .
Easy $A_{1}(0,-c)$ .

The line $CN \bot AC< :\ y=\frac{c-a}{b}(x-c)$ intersects the circle $(x-c)^{2}+y^{2}=(a-c)^{2}+b^{2}$ in the point $N(b+c,c-a)$ .
The midpoint of $AN$ is the midpoint of $CB_{1} \Rightarrow B_{1}(a+b,b+c-a)$ .

The lines $AA_{1}\ :\ y-b=\frac{b+c}{a}(x-a)$ and $BB_{1}\ :\ y=\frac{b+c-a}{a+b}x$ intersect in the point $S(\frac{ac(a+b)}{a^{2}+b^{2}+bc},\frac{ac(b+c-a)}{a^{2}+b^{2}+bc})$ .
The coordinates of $S$ satisfy the equation of $CC_{1}\ :\ y=-\frac{a}{b}(x-c)$ .

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