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Junior Balkan Mathematical Olympiad 2020- P2

Source: JBMO 2020

September 11, 2020
Juniorgeometrycyclic quadrilateralBalkan

Problem Statement

Let △ABC\triangle ABC be a right-angled triangle with ∠BAC=90∘\angle BAC = 90^{\circ} and let EE be the foot of the perpendicular from AA to BCBC. Let Z≠AZ \ne A be a point on the line ABAB with AB=BZAB = BZ. Let (c)(c) be the circumcircle of the triangle △AEZ\triangle AEZ. Let DD be the second point of intersection of (c)(c) with ZCZC and let FF be the antidiametric point of DD with respect to (c)(c). Let PP be the point of intersection of the lines FEFE and CZCZ. If the tangent to (c)(c) at ZZ meets PAPA at TT, prove that the points TT, EE, BB, ZZ are concyclic.
Proposed by Theoklitos Parayiou, Cyprus
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