Question 86
Let $M$ be an arbitrary point on the circumcircle of triangle $ABC$ and let the tangents from this point to the incircle of the triangle meet the sideline $BC$ at $X_1$, and $X_2$. Prove that the second intersection of the circumcircle of triangle $MX_1X_2$ with the circumcircle of $ABC$ (different from $M$) coincides with the tangency point of the circumcircle with mixtilinear incircle in angle $A$. (As usual, the A-mixtilinear incircle names the circle tangent to $AB, AC$ and to the circumcircle of $ABC$ internally.)