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Show that AB, CM, PQ are concurrent

Source: IMOC 2021 G6

August 11, 2021
geometrycircumcircleconcurrent

Problem Statement

Let Ω\Omega be the circumcircle of triangle ABCABC. Suppose that XX is a point on the segment ABAB with XB=XCXB=XC, and the angle bisector of ∠BAC\angle BAC intersects BCBC and Ω\Omega at DD, MM, respectively. If PP is a point on BCBC such that APAP is tangent to Ω\Omega and QQ is a point on DXDX such that CQCQ is tangent to Ω\Omega, show that ABAB, CMCM, PQPQ are concurrent.
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