The Dido problem with an arclength constraint

differential-geometry plane-curves euclidean-geometry

This diagram might help:

enter image description here

For any length $l > \overline{AB}$, we can find a circle passing through $AB$ such that the length of a circular arc between $A$ and $B$ is equal to $l$. In the diagram above, suppose that the red, dotted arc $ADB$ and the upper part of the circle both have length $l$. Note that the region bounded by $ADBC$ (i.e.: the pink area) has the same perimeter length as the circle.

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