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What are points P, Q doing here?

Source: Sharygin Correspondence Round 2024 P11

March 6, 2024
geometryperpendicular bisectorbarycentric coordinates

Problem Statement

Let M,NM, N be the midpoints of sides AB,ACAB, AC respectively of a triangle ABCABC. The perpendicular bisector to the bisectrix ALAL meets the bisectrixes of angles BB and CC at points PP and QQ respectively. Prove that the common point of lines PMPM and QNQN lies on the tangent to the circumcircle of ABCABC at AA.
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