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Median through point on circumcircle of square

Source: Sharygin Finals 2017, Problem 8.8

August 4, 2017
geometrysquareProjectivecircumcircle

Problem Statement

Let ABCDABCD be a square, and let PP be a point on the minor arc CDCD of its circumcircle. The lines PA,PBPA, PB meet the diagonals BD,ACBD, AC at points K,LK, L respectively. The points M,NM, N are the projections of K,LK, L respectively to CDCD, and QQ is the common point of lines KNKN and MLML. Prove that PQPQ bisects the segment ABAB.
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