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Turkey TST 2010 Q1

Source:

September 1, 2010
geometryincentercircumcircletrigonometrytrig identitiesLaw of Sinesgeometry proposed

Problem Statement

D, E, FD, \: E , \: F are points on the sides AB, BC, CA,AB, \: BC, \: CA, respectively, of a triangle ABCABC such that AD=AF, BD=BE,AD=AF, \: BD=BE, and DE=DF.DE=DF. Let II be the incenter of the triangle ABC,ABC, and let KK be the point of intersection of the line BIBI and the tangent line through AA to the circumcircle of the triangle ABI.ABI. Show that AK=EKAK=EK if AK=AD.AK=AD.
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