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a concyclicity with the foot of altitude

Source: Sharygin Final 2024 8.3

August 2, 2024
geometrySharygin Geometry OlympiadSharygin 2024

Problem Statement

Let ADAD be the altitude of an acute-angled triangle ABCABC and A′A' be the point on its circumcircle opposite to AA. A point PP lies on the segment ADAD, and points XX, YY lie on the segments ABAB, ACAC respectively in such a way that ∠CBP=∠ADY\angle CBP = \angle ADY, ∠BCP=∠ADX\angle BCP = \angle ADX. Let PA′PA' meet BCBC at point TT. Prove that DD, XX, YY, TT are concyclic.
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