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incircles of ABM,CBM are tangent , KA=AC=CN (2014 Kyiv City MO 11.4)

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August 6, 2020
geometryincircletangent circles

Problem Statement

In the triangle ABCABC, for which AC<AB<BCAC <AB <BC, on the sides ABAB and BCBC the points KK and NN were chosen, respectively, that KA=AC=CNKA = AC = CN. The lines ANAN and CKCK intersect at the point OO. From the point OO held the segment OM⊥ACOM \perp AC (M∈ACM \in AC) . Prove that the circles inscribed in triangles ABMABM and CBMCBM are tangent.
(Igor Nagel)
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