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\angle QXY = \angle ACP, \angle QYX = \angle ABP

Source: Iran MO Third Round 2021 G3

September 25, 2021
geometrycircumcircle

Problem Statement

Given triangle ABCABC variable points XX and YY are chosen on segments ABAB and ACAC, respectively. Point ZZ on line BCBC is chosen such that ZX=ZYZX=ZY. The circumcircle of XYZXYZ cuts the line BCBC for the second time at TT. Point PP is given on line XYXY such that ∠PTZ=90∘\angle PTZ = 90^ \circ. Point QQ is on the same side of line XYXY with AA furthermore ∠QXY=∠ACP\angle QXY = \angle ACP and ∠QYX=∠ABP\angle QYX = \angle ABP. Prove that the circumcircle of triangle QXYQXY passes through a fixed point (as XX and YY vary).
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