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

If A1A2...A2n (n >1) is any cyclic 2n-gon in which vertex Ai is connected to vertex Ai+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.

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } 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.

Hexagon investigations for students are available at Alternate Angles Sum Cyclic Hexagon and Alternate Sides Sum Circumscribed Hexagon.

Read my 1993 IJMEST paper discussing the above generalization, and its dual A unifying generalization of Turnbull's theorem and my 2006 Mathematics in School paper dealing with the converse Recycling cyclic polygons dynamically.

A neat application of this Angle Sum result for cyclic 2n-gons is the Angle Divider Theorem for a Cyclic Quadrilateral, and its further generalization to cyclic 2n-gons.

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.

Created by Michael de Villiers, 26 March 2012; updated to WebSketchpad, 31 August 2020.