Question 111
Let $ABC$ be a triangle with the shortest side $BC$. Let $X$, $Y$, $K$, $L$ be points on the sides $AB$, $AC$ and on the rays opposite to the rays $BC$, $CB$ respectively such that $BX = BK = BC = CY = CL$. The line $KX$ intersects the line $LY$ in a point $M$. Prove that the centroid of the triangle $KLM$ coincides with the incentre of the triangle $ABC$.
(Tomáš Jurík)
(Tomáš Jurík)