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Prove that four points are concyclic

Source: Indian IMOTC 2013, Team Selection Test 3, Problem 2

July 30, 2013
geometrycircumcircletrigonometryAsymptotegeometry proposed

Problem Statement

In a triangle ABCABC, let II denote its incenter. Points D,E,FD, E, F are chosen on the segments BC,CA,ABBC, CA, AB, respectively, such that BD+BF=ACBD + BF = AC and CD+CE=ABCD + CE = AB. The circumcircles of triangles AEF,BFD,CDEAEF, BFD, CDE intersect lines AI,BI,CIAI, BI, CI, respectively, at points K,L,MK, L, M (different from A,B,CA, B, C), respectively. Prove that K,L,M,IK, L, M, I are concyclic.
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