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if circumcenters of DEF,ABC coincide then ABC is equilateral

Source: Indian Postal Coaching 2008 set 6 p5

May 25, 2020
CircumcenterEquilateralratiogeometry

Problem Statement

Consider the triangle ABCABC and the points D∈(BC),E∈(CA),F∈(AB)D \in (BC),E \in (CA), F \in (AB), such that BDDC=CEEA=AFFB\frac{BD}{DC}=\frac{CE}{EA}=\frac{AF}{FB}. Prove that if the circumcenters of triangles DEFDEF and ABCABC coincide, then the triangle ABCABC is equilateral.
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