If A1A2...A2n (n >1) is any circumscribed 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 sides are equal.
(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.)
Drag any of the red tangent points in the sketches below to view the theorem dynamically.
A dual, circumscribed quadrilateral generalization of Euclid Book III, proposition 22
Note that in order for the theorem to work for n = 3, k = 2 (and for n = 4, k = 2; etc.) the concept 'side' needs to be interpreted (in the same way as for k = 1) as the sum of tangents from adjacent vertices. Unlike for k = 1 though, where these tangents meet at a common point and form a straight line, that is not the case for k = 2. Here a side very counter-intuitively has to be interpreted as A1R + A2Q with the corresponding alternate sides A3T + A4S; etc.
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.
Created by Michael de Villiers, 27 March 2012.