Problem

Source: Serbia JBMO TST 2022 P2

Tags: geometry, incenter, circumcircle, midpoints, mixtilinear incircle



Let $I$ be the incenter, $A_1$ and $B_1$ midpoints of sides $BC$ and $AC$ of a triangle $\Delta ABC$. Denote by $M$ and $N$ the midpoints of the arcs $AC$ and $BC$ of circumcircle of $\Delta ABC$ which do contain the other vertex of the triangle. If points $M$, $I$ and $N$ are collinear prove that: \begin{align*} \angle AIB_1=\angle BIA_1=90^{\circ} \end{align*}