A generalization of the Cyclic Quadrilateral Angle Sum theorem (Euclid Book III, proposition 22)

If A_{1}A_{2}...A_{2n} (n >1) is any cyclic 2n-gon in which vertex A_{i} is connected to vertex A_{i+k} and k = 1, 2, 3, ... n-1, then the two sums of alternate interior angles are each equal to (n-k)180°. (The value of k also corresponds to the total turning (number of complete revolutions) one would undergo walking around the perimeter, and turning at each vertex.)

The earliest mention and proof of the convex generalization for k = 1, seems to be that of Duncan Gregory who in 1836 published it in the Cambridge Mathematical Journal, Vol 1, p. 192. (Reference: Walton, William. (Ed). (1865). The Mathematical Writings of Duncan Gregory, Cambridge: Deighton, Bell & Co., p. 107. (download))

Drag any of the red or blue vertices of the cyclic polygons in the sketches below to view the theorem dynamically.

A generalization of the Cyclic Quadrilateral Angle Sum theorem

A similar Side Sum dual generalization to the above exists for Circumscribed/Tangential 2n-gons - a dynamic version is available at A circumscribed 2n-gon dual generalization.

A proof of this generalization and its dual (see link further up) is also given in Some Adventures in Euclidean Geometry, which is available for purchase as a downloadable PDF, printed book or from iTunes for your iPhone, iPad, or iPod touch, and on your computer with iTunes.