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given right & isosceles triange, another isosceles, 2 midpoints, square wanted

Source: Lusophon 2018 CPLP P2 , May Olympiad (Olimpiada de Mayo) 2002 L2

September 13, 2018
right triangleisoscelessquaregeometry

Problem Statement

In a triangle ABCABC, right in AA and isosceles, let DD be a point on the side ACAC (A≠D≠CA \ne D \ne C) and EE be the point on the extension of BABA such that the triangle ADEADE is isosceles. Let PP be the midpoint of segment BDBD, RR be the midpoint of the segment CECE and QQ the intersection point of EDED and BCBC. Prove that the quadrilateral ARQPARQP is a square.
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