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Tangent circles and concyclic points

Source: Romanian TST 3 2008, Problem 1

June 7, 2008
geometrycircumcirclegeometry proposed

Problem Statement

Let ABC ABC be a triangle with ∡BAC<∡ACB \measuredangle{BAC} < \measuredangle{ACB}. Let D D, E E be points on the sides AC AC and AB AB, such that the angles ACB ACB and BED BED are congruent. If F F lies in the interior of the quadrilateral BCDE BCDE such that the circumcircle of triangle BCF BCF is tangent to the circumcircle of DEF DEF and the circumcircle of BEF BEF is tangent to the circumcircle of CDF CDF, prove that the points A A, C C, E E, F F are concyclic. Author: Cosmin Pohoata
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