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Tangents to cirumcircle meet at the fixed point T

Source: Viet Nam TST 2012 Day 1

April 17, 2012
geometrycircumcirclesearchgeometric transformationreflectiongeometry unsolved

Problem Statement

Consider a circle (O)(O) and two fixed points B,CB,C on (O)(O) such that BCBC is not the diameter of (O)(O). AA is an arbitrary point on (O)(O), distinct from B,CB,C. Let D,J,KD,J,K be the midpoints of BC,CA,ABBC,CA,AB, respectively, E,M,NE,M,N be the feet of perpendiculars from AA to BCBC, BB to DJDJ, CC to DKDK, respectively. The two tangents at M,NM,N to the circumcircle of triangle EMNEMN meet at TT. Prove that TT is a fixed point (as AA moves on (O)(O)).
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