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

If A_{1}A_{2}...A_{2n} (n >1) is any circumscribed 2n-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 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 A_{1}R + A_{2}Q with the corresponding alternate sides A_{3}T + A_{4}S; etc.

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.