A dual, circumscribed quadrilateral generalization of Euclid Book III, proposition 22

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.)

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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.

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Created by Michael de Villiers, 27 March 2012.