Prove that MN ext is parallel to AP.

AB is the diameter of the circle, so the angle subtended by the diameter at any point on the circumference is $90^\circ$. Therefore, $\angle APB = 90^\circ$, which means $AP \perp PB$.
We are given that $MN$ is perpendicular to $PB$. So, $MN \perp PB$.

Since both line segment $AP$ and line segment $MN$ are perpendicular to the same line segment $PB$, they must be parallel to each other.
Therefore, $MN \parallel AP$.

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