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<BLC=2<BAC wanted, symmedian and circumcircle related

Source: Champions Tournament (Ukraine) - Турнір чемпіонів - 2005 Seniors p2

September 6, 2020
geometrycircumcirclesymmediananglesequal anglesChampions Tournament

Problem Statement

Given a triangle ABCABC, the line passing through the vertex AA symmetric to the median AMAM wrt the line containing the bisector of the angle ∠BAC\angle BAC intersects the circle circumscribed around the triangle ABCABC at points AA and KK. Let LL be the midpoint of the segment AKAK. Prove that ∠BLC=2∠BAC\angle BLC=2\angle BAC.
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