collinear, 2 squares inscribed in triangle (2019-20 Savin Competition 7-9 p22)

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collinear, 2 squares inscribed in triangle (2019-20 Savin Competition 7-9 p22)

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#1 h Jun 24, 2021, 10:19 PM

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Two squares are inscribed in a right-angled triangle, as shown in the figure. Prove that three marked points lie on one straight line.

https://cdn.artofproblemsolving.com/attachments/d/7/703d793df6bcd0747a9cce0146e52c9a4b4eb3.png

This post has been edited 2 times. Last edited by parmenides51, Jun 25, 2021, 3:59 AM

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#2 h Jun 24, 2021, 10:20 PM

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#3 h Jun 25, 2021, 2:41 PM • 2 Y

Y by Mango247, Mango247

Given $A(0,0),B(b,0),C(0,c)$ .

Point $D(\frac{bc}{b+c},0)$ on the line $AB$ , the blue square.
Point $E(0,\mu)$ on the line $AC$ , the red square; then $F(\frac{b\mu}{c},0)$ on the line $AB$ .

Distance $EF$ must be equal to the distance of $E$ to the line $BC$ , giving $\mu=\frac{bc^{2}}{b^{2}+bc+c^{2}}$ .

The squares cut in the point $(\frac{bc}{b+c},\frac{b^{2}c^{2}}{(b+c)(b^{2}+bc+c^{2})}\ )$ .

The points $B(b,0),(0,\frac{bc^{2}}{b^{2}+bc+c^{2}}$ and $(\frac{bc}{b+c},\frac{b^{2}c^{2}}{(b+c)(b^{2}+bc+c^{2})}\ )$ lie on the line $y=-\frac{c^{2}}{b^{2}+bc+c^{2}}(x-b)$ .

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