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Prove line tangent to circle

Source: China Mathematical Olympiad 2019 Q3

November 14, 2018
geometrycircumcircle

Problem Statement

Let OO be the circumcenter of △ABC\triangle ABC(AB<ACAB<AC), and DD be a point on the internal angle bisector of ∠BAC\angle BAC. Point EE lies on BCBC, satisfying OE∥ADOE\parallel AD, DE⊥BCDE\perp BC. Point KK lies on EBEB extended such that EK=EAEK=EA. The circumcircle of △ADK\triangle ADK meets BCBC at P≠KP\neq K, and meets the circumcircle of △ABC\triangle ABC at Q≠AQ\neq A. Prove that PQPQ is tangent to the circumcircle of △ABC\triangle ABC.
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