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IMAC 2010 seniors, first day, third problem.

Source:

July 5, 2010
geometrytrigonometrygeometry proposed

Problem Statement

Let ABCABC be a triangle and let D∈(BC)D\in (BC) be the foot of the AA- altitude. The circle ww with the diameter [AD][AD]
meet again the lines ABAB , ACAC in the points K∈(AB)K\in (AB) , L∈(AC)L\in (AC) respectively. Denote the meetpoint MM
of the tangents to the circle ww in the points KK , LL . Prove that the ray [AM[AM is the AA-median in △ABC\triangle ABC (Serbia).
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