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Geometry problem

Source: 2017 Taiwan TST

April 13, 2018
geometry

Problem Statement

Given a circle and four points B,C,X,YB,C,X,Y on it. Assume AA is the midpoint of BCBC, and ZZ is the midpoint of XYXY. Let L1,L2L_1,L_2 be lines perpendicular to BCBC and pass through B,CB,C respectively. Let the line pass through XX and perpendicular to AXAX intersects L1,L2L_1,L_2 at X1,X2X_1,X_2 respectively. Similarly, let the line pass through YY and perpendicular to AYAY intersects L1,L2L_1,L_2 at Y1,Y2Y_1,Y_2 respectively. Assume X1Y2X_1Y_2 intersects X2Y1X_2Y_1 at PP. Prove that ∠AZP=90o.\angle AZP=90^o.
Proposed by William Chao
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