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equal angles wanted, circles, tangents, symmetric point given

Source: Ukrainian Geometry Olympiad 2020, IX p2, X p1, XI p1

April 27, 2020
geometryequal anglescirclestangentsymmetry

Problem Statement

Let Γ\Gamma be a circle and PP be a point outside, PAPA and PBPB be tangents to Γ\Gamma , A,B∈ΓA, B \in \Gamma . Point KK is an arbitrary point on the segment ABAB. The circumscirbed circle of △PKB\vartriangle PKB intersects Γ\Gamma for the second time at point TT, point P′P' is symmetric to point PP wrt point AA. Prove that ∠PBT=∠P′KA\angle PBT = \angle P'KA.
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