Miquel’s pentagon theorem

An earlier post presented an elegant plane geometry theorem discovered by the 19th century school teacher Auguste Miquel. This post presents his pentagon theorem. Start with a pentagon. It may be irregular, but it needs to be convex. Extend each of the sides of the pentagon to form a star, then draw give circles, one through each of the triangles formed by a side of the pentagon and a vertex of the star. The five circles intersect in pairs at ten points: the five verices of the pentagon and five new points. The five new points lie on a circle. The converse of this theorem is known as the five circles theorem.The post Miquel’s pentagon theorem first appeared on John D. Cook.