Dido's Problem

Classic Problem
The new is to be founded.

Its area is to be determined by enclosing as much land as possible within a line whose length is fixed.

What shape should it be to make sure the area is a maximum?

Solution
The classical answer is that the enclosure was in the shape of a semicircle whose diameter is the coastline.

However, it is clear that this depends upon the shape of the coast, and a better solution may be to cut off a.

Proof
Let the shoreline be assumed to be a straight line.

Imagine the enclosure takes some geometric figure $S$.

Let $S$ be reflected in the shoreline.

Then the entire geometric figure formed by $S$ along with its reflection $S'$ encloses the largest area for double the length of the boundary line.

This largest area is a circle.