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Prove that there exists a point Q

Source: Hungary-Israel Binational Olympiad 2009, Problem 4

August 17, 2009
geometrytrapezoidcircumcircleparallelogrampower of a pointradical axiscomplex numbers

Problem Statement

Given is the convex quadrilateral ABCD ABCD. Assume that there exists a point P P inside the quadrilateral for which the triangles ABP ABP and CDP CDP are both isosceles right triangles with the right angle at the common vertex P P. Prove that there exists a point Q Q for which the triangles BCQ BCQ and ADQ ADQ are also isosceles right triangles with the right angle at the common vertex Q Q.