||Exploring bicentric polygons
A bicentric polygon is a polygon which is simultaneously cyclic: all vertices lie on a single circle, and tangential: all edges are tangential to another circle. All triangles are bicentric, as are all regular polygons. However, non-regular bicentric polygons of orders four or higher have some interesting properties, and they have been the subject of study and interest since the time of Euler. In particular the radii of the two circles, and of the distance between their centres, satisfy formulas which get progressively more complex as the number of sides increases. In this article we show how such formulas—and some fine diagrams—can be obtained with dynamic geometry software. This brings a somewhat niche area of geometry into the realm of modern experimentation, and what has previously required some very complicated algebra into the reach of an able school student.