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Circumcenters coincide

Source: LXII Polish Olympiad 2011, Problem 2

May 24, 2011
geometrycircumcircleincentergeometric transformationdilationperpendicular bisectorpower of a point

Problem Statement

The incircle of triangle ABCABC is tangent to BC,CA,ABBC,CA,AB at D,E,FD,E,F respectively. Consider the triangle formed by the line joining the midpoints of AE,AFAE,AF, the line joining the midpoints of BF,BDBF,BD, and the line joining the midpoints of CD,CECD,CE. Prove that the circumcenter of this triangle coincides with the circumcenter of triangle ABCABC.
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