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Random point on circumcircle makes a equal angle made with reflections

Source: Azerbaijan IMO TST 2022, D1 P3

August 23, 2023
geometryAZE IMO TST

Problem Statement

Let ABCABC be a triangle with circumcircle ω\omega and DD be any point on ω.\omega. Suppose that PP is the midpoint of chord ADAD and points X,YX, Y are chosen on lines AC,ABAC, AB such that reflections of B,CB, C with respect to ADAD lie on XP,YP,XP, YP, respectively. If the circumcircle of triangle AXYAXY intersects ω\omega at II for the second time, prove that ∠PID\angle PID equals the angle formed by lines ADAD and BC.BC. Proposed by tenplusten.
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