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centroid is incenter of other triangle, (2019 Kyiv City MO Round2 11.3.1)

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September 19, 2020
geometryincenterCentroidequal segments

Problem Statement

It is known that in the triangle ABCABC the smallest side is BCBC. Let X,Y,KX, Y, K and LL - points on the sides AB,ACAB, AC and on the rays CB,BCCB, BC, respectively, are such that BX=BK=BC=CY=CLBX = BK = BC =CY =CL. The line KXKX intersects the line LYLY at the point MM. Prove that the intersection point of the medians △KLM\vartriangle KLM coincides with the center of the inscribed circle △ABC\vartriangle ABC.
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