Equable Shape - Cyclic Polygons

Cyclic Polygons

A cyclic polygon is a polygon in which the sides are all tangent to a common circle. Every cyclic polygon may be triangulated by drawing edges from the circle's center to its vertices, forming a collection of triangles that all have height equal to the radius; it follows from this decomposition that the total area of a cyclic polygon equals half the radius times the perimeter. Thus, a cyclic polygon is equable if and only if its inradius is two. All triangles are cyclic, so in particular the equable triangles are exactly the triangles with inradius two.