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Three collinear points

Source: Iranian third round geometry problem 4

September 10, 2015
geometryincenter

Problem Statement

Let ABCABC be a triangle with incenter II. Let KK be the midpoint of AIAI and BI∩⊙(△ABC)=M,CI∩⊙(△ABC)=NBI\cap \odot(\triangle ABC)=M,CI\cap \odot(\triangle ABC)=N. points P,QP,Q lie on AM,ANAM,AN respectively such that ∠ABK=∠PBC,∠ACK=∠QCB\angle ABK=\angle PBC,\angle ACK=\angle QCB. Prove that P,Q,IP,Q,I are collinear.
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