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Circumcenters of triangles coincide

Source: RMO 2008, Grade 10, Problem 1

April 30, 2008
geometrycircumcirclegeometric transformationreflectionperpendicular bisectorgeometry proposed

Problem Statement

Let ABC ABC be a triangle and the points D∈(BC) D\in (BC), E∈(CA) E\in (CA), F∈(AB) F\in (AB) such that \frac {BD}{DC} \equal{} \frac {CE}{EA} \equal{} \frac {AF}{FB}. Prove that if the circumcenters of the triangles DEF DEF and ABC ABC coincide then ABC ABC is equilateral.
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