Robbins pentagon

Buchholz and MacDougall chose this name by analogy with the naming of Heron triangles after Hero of Alexandria, the discoverer of Heron's formula for the area of a triangle as a function of its edge lengths.

Buchholz and MacDougall also showed that, in every Robbins pentagon, either all five of the internal diagonals are rational numbers or none of them are.

If the five diagonals are rational (the case called a Brahmagupta pentagon by Sastry (2005)), then the radius of its circumscribed circle must also be rational, and the pentagon may be partitioned into three Heronian triangles by cutting it along any two non-crossing diagonals, or into five Heronian triangles by cutting it along the five radii from the circle center to its vertices.

Buchholz and MacDougall performed computational searches for Robbins pentagons with irrational diagonals but were unable to find any.

On the basis of this negative result they suggested that Robbins pentagons with irrational diagonals may not exist.