Regiomontanus' Angle Maximization Problem
Let $AB$ be a line segment.
Let $AB$ be produced to $P$.
Let $PQ$ be constructed perpendicular to $AB$.
All other points on $PQ$ that are not $Q$ itself are outside $C$.
Hence the result.
Source of Name
This entry was named for Regiomontanus.
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.