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CR = 2XY wanted, AQ = AC, AR = CP, circumcircle realted

Source: 2016 Thailand October Camp 1.3

October 15, 2020
geometrycircumcircleequal segments

Problem Statement

In △ABC\vartriangle ABC with AB>ACAB > AC, the tangent to the circumcircle at AA intersects line BCBC at PP. Let QQ be the point on ABAB such that AQ=ACAQ = AC, and AA lies between BB and QQ. Let RR be the point on ray APAP such that AR=CPAR = CP. Let X,YX, Y be the midpoints of AP,CQAP, CQ respectively. Prove that CR=2XYCR = 2XY .
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