Question 83
Let $ABCD$ be a convex quadrilateral inscribed in a circle $(O)$. Suppose that the line $AB$ meets the line $CD$ at $M$ and the line $AD$ meets the line $BC$ at $N$. Let $P, Q, S$, and $T$ be the intersection points of the bisectors of the angles $\widehat{MAN}$ and $\widehat{MBN}$, $\widehat{MBN}$ and $\widehat{MCN}$, $\widehat{MCN}$ and $\widehat{MDN}$, $\widehat{MDN}$ and $\widehat{MAN}$, respectively. Suppose that $P, Q, S$ and $T$ are pairwise different.
1. Prove that four points $P, Q, S$ and $T$ are cyclic. Let $I$ be the center of the cycle $(P, Q, S, T)$.
2. Let $E$ be the intersection of $AC$ and $BD$. Prove that three points $E, O, I$ are collinear.
1. Prove that four points $P, Q, S$ and $T$ are cyclic. Let $I$ be the center of the cycle $(P, Q, S, T)$.
2. Let $E$ be the intersection of $AC$ and $BD$. Prove that three points $E, O, I$ are collinear.