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

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))

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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 2n-gons - a dynamic version is available at A circumscribed 2n-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.