Problem

Source: 2021 Saudi Arabia BMO TST 2.2

Tags: geometry, parallel, circumcircle



Let $ABC$ be an acute triangle with $AB < AC$ and inscribed in the circle $(O)$. Denote $I$ as the incenter of $ABC$ and $D$, $E$ as the intersections of $AI$ with $BC$, $(O)$ respectively. Take a point $K$ on $BC$ such that $\angle AIK = 90^o$ and $KA$, $KE$ meet $(O)$ again at M,N respectively. The rays $ND$, $NI$ meet the circle $(O)$ at $Q$,$P$. 1. Prove that the quadrilateral $MPQE$ is a kite. 2. Take $J$ on $IO$ such that $AK \perp AJ$. The line through $I$ and perpendicular to $OI$ cuts $BC$ at $R$ ,cuts $EK$ at $S$ .Prove that $OR \parallel JS$.