The hexaparagon

Abstract

A hexagon with each pair of opposite sides parallel to a side of a triangle will be called a hexaparagon for that triangle. One way to construct a hexaparagon for a given triangle ABC is to use as vertices the centroids P, Q, R, S, T, and U of the six non-overlapping sub-triangles formed by the three medians of triangle ABC. The perimeter of this hexaparagon is half the perimeter of triangle ABC. The ratio of the areas of triangle ABC to this hexaparagon is 36 to 13 and the lengths of the parallel sides are in the ratio 6 to 2 to 1. The vertices of this hexaparagon lie on an ellipse and, with a second type of hexaparagon introduced later, hexaparagons tile the plane.