Elementary proof of a theorem about hexagons with parallel opposite sides

Abstract

The perpendicular bisectors of the sides of a hexagon, whose opposite sides are parallel, produce a point symmetric hexagon. Michael de Villiers already gave two proofs of this theorem, firstly an elaborate one with the aid of dynamic geometry and secondly a merely verifying one with the help of coordinates and computer algebra. In this note, we present a short elementary proof. Our proof uses the fact, that the three altitudes of a triangle intersect in a common point, and it also covers degenerate cases.