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PAMO Problem 3: Circumcentres coincide implies equilateral

Source: 2019 Pan-African Mathematics Olympiad, Problem 3

April 9, 2019
geometryCircumcenterPAMOEquilateral Trianglecircumcircle

Problem Statement

Let ABCABC be a triangle, and DD, EE, FF points on the segments BCBC, CACA, and ABAB respectively such that BDDC=CEEA=AFFB. \frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}. Show that if the centres of the circumscribed circles of the triangles DEFDEF and ABCABC coincide, then ABCABC is an equilateral triangle.
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