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An isosceles triangle is given, prove a right angle

Source: ISL 2020 G1

July 20, 2021
geometryright triangleIsosceles TriangleIMO ShortlistIMO Shortlist 2020

Problem Statement

Let ABCABC be an isosceles triangle with BC=CABC=CA, and let DD be a point inside side ABAB such that AD<DBAD< DB. Let PP and QQ be two points inside sides BCBC and CACA, respectively, such that ∠DPB=∠DQA=90∘\angle DPB = \angle DQA = 90^{\circ}. Let the perpendicular bisector of PQPQ meet line segment CQCQ at EE, and let the circumcircles of triangles ABCABC and CPQCPQ meet again at point FF, different from CC. Suppose that PP, EE, FF are collinear. Prove that ∠ACB=90∘\angle ACB = 90^{\circ}.
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