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, 2015

Geometry Difficulty 6.0 National Olympiad Prove it Hungary

In the acute angled triangle $ABC$ , the midpoints of the sides $BC$ , $CA$ and $AB$ are $D$ , $E$ and $F$ , respectively. The foot of the altitude of the triangle starting from $C$ is $T_1$ . On some line, passing through point $C$ but not containing $T_1$ , the feet of the perpendiculars starting from $A$ and $B$ are $T_2$ and $T_3$ , respectively. Prove that the circle $DEF$ passes through the center of the circle $T_1T_2T_3$ .
Proposed by: Bálint Bíró, Eger
(5 pont)

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