Question 145
Let $ABC$ be a non-isosceles triangle inscribed in a circle $(O)$ and $BE$, $CF$ are two angle bisectors intersecting at $I$ with $E$ belonging to segment $AC$ and $F$ belonging to segment $AB$. Suppose that $BE$, $CF$ intersect $(O)$ at $M$, $N$ respectively. The line $d_{1}$ passes through $M$ and is perpendicular to $BM$, intersecting $(O)$ at the second point $P$; the line $d_{2}$ passes through $N$ and is perpendicular to $CN$, intersecting $(O)$ at the second point $Q$. Denote $H$, $K$ as the midpoints of $MP$ and $NQ$ respectively.
1. Prove that triangles $IEF$ and $OKH$ are similar.
2. Suppose that $S$ is the intersection of the two lines $d_{1}$ and $d_{2}$. Prove that $SO$ is perpendicular to $EF$.
1. Prove that triangles $IEF$ and $OKH$ are similar.
2. Suppose that $S$ is the intersection of the two lines $d_{1}$ and $d_{2}$. Prove that $SO$ is perpendicular to $EF$.