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similar isosceles triangles on sides of a triangle, # or colinear

Source: Polish MO Recond Round 1984 p2

September 9, 2024
geometrysimilar trianglesparallelogram

Problem Statement

We construct similar isosceles triangles on the sides of the triangle ABC ABC : triangle APB APB outside the triangle ABC ABC (AP=PB AP = PB ), triangle CQA CQA outside the triangle ABC ABC (CQ=QA CQ = QA ), triangle CRB CRB inside the triangle ABC ABC (CR=RB CR = RB ). Prove that APRQ APRQ is a parallelogram or that the points A,P,R,Q A, P, R, Q lie on a straight line.
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