Question 47
Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$ and let $M$ be the midpoint of $BC$. The bisector of angle $A$ intersects lines $BC$ and $OM$ at $L$ and $Q$, respectively. Prove that
$$
AI \cdot LQ = IL \cdot IQ.
$$
$$
AI \cdot LQ = IL \cdot IQ.
$$