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b+c=2a Make Cyclic Quadrilateral

Source: Iran Third Round 2013 - Geometry Exam - Problem 2

September 7, 2013
geometrycircumcircleincentergeometric transformationgeometry unsolved

Problem Statement

Let ABCABC be a triangle with circumcircle (O)(O). Let M,NM,N be the midpoint of arc AB,ACAB,AC which does not contain C,BC,B and let M′,N′M',N' be the point of tangency of incircle of △ABC\triangle ABC with AB,ACAB,AC. Suppose that X,YX,Y are foot of perpendicular of AA to MM′,NN′MM',NN'. If II is the incenter of △ABC\triangle ABC then prove that quadrilateral AXIYAXIY is cyclic if and only if b+c=2ab+c=2a.
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