Question 6
Let $ABC$ be an acute triangle with $AB \lt AC$ 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 $\angle AIK = 90^\circ$ and $KA, KE$ meet $(O)$ again at $M, N$ respectively. The rays $ND, NI$ meet the circle $(O)$ at $Q, P$.
**Part 1.** Prove that the quadrilateral $MPQE$ is a kite.
**Part 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 $OR \parallel JS$.
**Part 1.** Prove that the quadrilateral $MPQE$ is a kite.
**Part 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 $OR \parallel JS$.