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Prove that line $QM$ passes through the midpoint of $AC.$

Source: Moldova TST 2023

April 9, 2023
geometry

Problem Statement

Let ABCABC be a triangle with ∠ACB=90\angle ACB=90 and AC>BC.AC>BC. Let Ω\Omega be the circumcircle of ABC.ABC. Point D D is the midpoint of small arc ACAC of Ω.\Omega. Point M M is symmetric with A A with respect to D.D. Point N N is the midpoint of MC.MC. Line ANAN intersects Ω\Omega in point P P and line BPBP intersects line DNDN in point Q.Q. Prove that line QMQM passes through the midpoint of AC.AC.
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