Problem

Source: Iranian TST 2020, second exam day 2, problem 4

Tags: geometry, incenter, circumcircle, Harmonics, homothety, Hi



Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent. Proposed by Alireza Dadgarnia