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Circumcentre is Incentre

Source: China Mathematical Olympiad 2014

December 21, 2013
geometryincenterratiosymmetrycircumcircletrigonometryangle bisector

Problem Statement

Let ABCABC be a triangle with AB>ACAB>AC. Let DD be the foot of the internal angle bisector of AA. Points FF and EE are on AC,ABAC,AB respectively such that B,C,F,EB,C,F,E are concyclic. Prove that the circumcentre of DEFDEF is the incentre of ABCABC if and only if BE+CF=BCBE+CF=BC.
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