Regiomontanus' Angle Maximization Problem

Theorem
Let $AB$ be a line segment.

Let $AB$ be produced to $P$.

Let $PQ$ be constructed perpendicular to $AB$.

Then the angle $AQB$ is greatest when $PQ$ is tangent to a circle passing through $A$, $B$ and $Q$.

Proof

 * Regiomontanus-angle-min-problem.png

There exists a unique circle $C$ tangent to $PQ$ passing through $A$ and $B$.

From Angles in Same Segment of Circle are Equal, the angle subtended by $AB$ from any point on $C$ is the same as angle $\angle AQB$.

All other points on $PQ$ that are not $Q$ itself are outside $C$.

Hence the angle subtended by $AB$ to such points is smaller.

Hence the result.