# 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

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.

$\blacksquare$

## Source of Name

This entry was named for Regiomontanus.

## Historical Note

Regiomontanus first posed this problem in $1471$ to Christian Roder.

He posed it in the context of choosing the optimum place to stand to view a statue positioned above eye level.

Too close and it will appear heavily foreshortened, too far away and it will just appear small.

It is notable as the first extremal problem since Heron's Principle of Reflection.

The problem has been reinvented several times, and has a contemporary application: it gives the best place to take a conversion kick in the game of rugby.

## Sources

- 1991: David Wells:
*Curious and Interesting Geometry*... (previous) ... (next):**angle in the same segment** - 1992: David Wells:
*Curious and Interesting Puzzles*... (previous) ... (next): The Best View of a Statue: $93$