$\omega$ is incircle of $\triangle ABC$ touch $AC$ in $S$. Point $Q$ lies on $\omega$ and midpoints of $AQ$ and $QC$ lies on $\omega$ . Prove that $QS$ bisects $\angle AQC$
Source: Moscow Math Olympiad 2017, Grade 11, P2
Tags: geometry
$\omega$ is incircle of $\triangle ABC$ touch $AC$ in $S$. Point $Q$ lies on $\omega$ and midpoints of $AQ$ and $QC$ lies on $\omega$ . Prove that $QS$ bisects $\angle AQC$