Question 13
Given a circle $C$ and a point $P$ outside it, the two tangents to the circle are drawn through $P$, with $A$ and $B$ being the points of tangency.
A point $Q$ is taken on the minor arc $AB$ of $C$. Let $M$ be the intersection of the line $AQ$ with the perpendicular to $AQ$ drawn by $P$ and let $N$ be the intersection of the line $BQ$ with the perpendicular to $BQ$ drawn by $P$.
Show that, by varying $Q$ in the arc $AB$, all the lines $MN$ pass through the same point.
A point $Q$ is taken on the minor arc $AB$ of $C$. Let $M$ be the intersection of the line $AQ$ with the perpendicular to $AQ$ drawn by $P$ and let $N$ be the intersection of the line $BQ$ with the perpendicular to $BQ$ drawn by $P$.
Show that, by varying $Q$ in the arc $AB$, all the lines $MN$ pass through the same point.