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Concyclic Points in an Isosceles Triangle

Source: Turkey TST 2016 P6

April 10, 2016
geometry

Problem Statement

In a triangle ABCABC with AB=ACAB=AC, let DD be the midpoint of [BC][BC]. A line passing through DD intersects ABAB at KK, ACAC at LL. A point EE on [BC][BC] different from DD, and a point PP on AEAE is taken such that ∠KPL=90∘−12∠KAL\angle KPL=90^\circ-\frac{1}{2}\angle KAL and EE lies between AA and PP. The circumcircle of triangle PDEPDE intersects PKPK at point XX, PLPL at point YY for the second time. Lines DXDX and ABAB intersect at MM, and lines DYDY and ACAC intersect at NN. Prove that the points P,M,A,NP,M,A,N are concyclic.
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