Area of a Lune
A lune is a plane figure bounded by the arcs of two circles. As a planar figure, the lune has an area but determining the area may be a challenge.
It is well known that a square with the same area as a circle can not be constructed with straightedge and compass. Surprisingly, even though the lune is bounded by arcs of circles, some lunar shapes can be shown to have a constructable square of the same area.
Given a right triangle ABC. Construct a lune with vertices at A and C with the center of the inner arc being at C and the center of the outer arc being at the midpoint of segment AB.
Prove that the area of the lune so constructed is the same as the area of the triangle ABC.
Hints: Some auxillary figures that may be helpful.
This is sometimes described as a "lune formed on the side of a square" and is associated with Hippocrates, who discovered a construction that showed a square could be constructed using straightedge and compass with the same area as this lune.
It is, of course, a special Lune. There are exactly 5 lunes for which a construction can be found to create a square of the same area using straightedge and compass.