## A generalization of Euclid Book III, proposition 22 (cyclic quadrilateral theorem) and its dual

If *A*_{1}A_{2}...A_{2n} (*n* >1) is any cyclic 2*n*-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 vertices in the sketches below to view the theorem dynamically.

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A generalization of Euclid Book III, proposition 22 (cyclic quadrilateral theorem)

A similar dual generalization to the above exists for circumscribed 2*n*-gons - a dynamic version is available at A circumscribed 2*n*-gon dual generalization.

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 A unifying generalization of Turnbull's theorem and my 2006 *Mathematics in School* paper dealing with the converse Recycling cyclic polygons dynamically.

A proof of this result 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.

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Created by Michael de Villiers, 26 March 2012.