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Isosceles triangle geo with parallel lines and tangencies

Source: IZHO 2023 P2

February 6, 2023
geometry

Problem Statement

The tangent at CC to Ω\Omega, the circumcircle of scalene triangle ABCABC intersects ABAB at DD. Through point DD, a line is drawn that intersects segments ACAC and BCBC at KK and LL respectively. On the segment ABAB points MM and NN are marked such that AC∥NLAC \parallel NL and BC∥KMBC \parallel KM. Lines NLNL and KMKM intersect at point PP lying inside the triangle ABCABC. Let ω\omega be the circumcircle of MNPMNP. Suppose CPCP intersects ω\omega again at QQ. Show that DQDQ is tangent to ω\omega.
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