Question 131
A quadrilateral $ABCD$ Is inscribed into a circle $\omega$. Its Diagonals Intersect at Point $K$. Let $M_1, M_2, M_3, M_4$ Be the Midpoints of Arcs $AB$, $BC$, $CD$, $DA$ (Not Containing the Other Vertices of $ABCD$), Respectively. Let $I_1, I_2, I_3, I_4$ Be the Incenters of Triangles $ABK$, $BCK$, $CDK$, $DAK$, respectively. Prove that the lines $M_1I_1$, $M_2I_2$, $M_3I_3$, $M_4I_4$ are concurrent. (P. Kozhevnikov) Quadrangle $ABCD$ Inscribed in a circle $\omega$, and its diagonals intersect at the point $K$. points $M_1, M_2, M_3, M_4$ - the middle of the arcs $AB$, $BC$, $CD$, $DA$ (not containing other vertices of the quadrilateral) respectively. points $I_1, I_2, I_3, I_4$ the centers of circles inscribed in triangles $ABK$, $BCK$, $CDK$, $DAK$ respectively. Prove that straight lines $M_1I_1$, $M_2I_2$, $M_3I_3$, $M_4I_4$ intersect at one point. (P. Kozhevnikov)