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Bosnia and Herzegovina TST 2012 Problem 5

Source:

May 20, 2012
geometrycircumcircletrigonometryTriangle

Problem Statement

Given is a triangle △ABC\triangle ABC and points MM and KK on lines ABAB and CBCB such that AM=AC=CKAM=AC=CK. Prove that the length of the radius of the circumcircle of triangle △BKM\triangle BKM is equal to the lenght OIOI, where OO and II are centers of the circumcircle and the incircle of △ABC\triangle ABC, respectively. Also prove that OI⊥MKOI\perp MK.
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