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

Generalization: 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.)

Historical Background: 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 Gregory paper))

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

(Note: For simplicity's sake, in the illustrative dynamic sketches above, it is assumed that the vertices Ai are cyclically arranged in order around the circle.)

Dual Generalization: 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 Cyclic Hexagon Alternate Angles Sum Theorem and Circumscribed Hexagon Alternate Sides Sum Theorem.

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

Application: 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.

Other Proofs: 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; updated 24 June 2023.