Question 3
Two circles $(O_1),(O_2)$ intersect at $A,B$. Their tangents to $(O_1)$ at $A,B$ meet at $K$. A point $M\ne A,B$ lies on $(O_1)$. Let $P$ be the second intersection of $MA$ with $(O_2)$, $C$ the second intersection of $MK$ with $(O_1)$, and $Q$ the second intersection of $CA$ with $(O_2)$. Prove:
1. The midpoint of $PQ$ lies on $MC$.
2. The line $PQ$ passes through a fixed point as $M$ varies.
As usual, a construction is considered only where the named second intersections and the lines in the claim are defined and nondegenerate.
1. The midpoint of $PQ$ lies on $MC$.
2. The line $PQ$ passes through a fixed point as $M$ varies.
As usual, a construction is considered only where the named second intersections and the lines in the claim are defined and nondegenerate.