Some Poncelet invariants for bicentric hexagons

Abstract

Tangential polygons are (convex) polygons for which every side is tangent to an inscribed circle. Cyclic polygons are those for which every vertex lies on a circle, the circumcircle. Bicentric $n$-gons are those which are both tangential and cyclic. Every triangle is bicentric. Bicentric quadrilaterals are those for which the sum of the lengths of opposite sides is the semiperimeter and for which opposite angles sum to $\pi$. Here we give some results pertaining to invariants of (convex) bicentric hexagons.

A remarkable result of Poncelet is that if one has a pair of circles admitting a bicentric $n$-gon, then for every point on the circumcircle can be a vertex for a bicentric $n$-gon. This is illustrated in the animation at\\ \verb$https://mathworld.wolfram.com/PonceletsPorism.html$

The animation indicates that, along with the incentre and circumcentre, the point of intersection of the principal diagonals of a $2m$-gon is invariant under the motion. Such invariants -- here called Poncelet invariants -- have been studied for two centuries, in particular for triangles and bicentric quadrilaterals. We present results, for bicentric hexagons, that various combinations of distances between vertices - lengths of diagonals and of sides - are invariant.