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square , 4 right isosceles triangles -VI Soros Olympiad 1999-00 Round 1 9.9

Source:

May 21, 2024
geometrysquare

Problem Statement

On the plane there are two isosceles non-intersecting right triangles ABCABC and DECDEC (ABAB and DEDE are the hypotenuses,ABDE ABDE is a convex quadrilateral), and AB=2DEAB = 2 DE. Let's construct two more isosceles right triangles: BDFBDF (with hypotenuse BFBF located outside triangle BDCBDC) and AEGAEG (with hypotenuse AGAG located outside triangle AECAEC). Prove that the line FGFG passes through a point NN such that DCENDCEN is a square.