The Tangential (or Circumscribed) Polygon Side Sum theorem

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.)

The theorem for n = 2 and k = 1 is called Pitot's Theorem, and named after French engineer Henri Pitot (1695-1771) who proved the forward implication in 1725. The converse was proved by the Swiss mathematician Jakob Steiner (1796-1863) in 1846.

A similar Angle Sum dual generalization to the above exists for Cyclic 2n-gons - a dynamic version is available at A Cyclic 2n-gon dual generalization.

Drag any of the red tangent points in the sketches below to view the theorem dynamically.

.sketch_canvas { border: medium solid lightgray; display: inline-block; } The Tangential (or Circumscribed) Polygon Side Sum theorem

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.

A similar Angle Sum dual generalization to the above exists for Cyclic 2n-gons - a dynamic version is available at A cyclic 2n-gon dual generalization.

Hexagon investigations for students are available at Cyclic Hexagon Alternate Angles Sum theorem and Circumscribed Hexagon Alternate Sides Sum theorem.

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

A neat application of this Side Sum result for circumscribed 2n-gons is the Side Divider Theorem for a Circumscribed Quadrilateral, and its further generalization to circumscribed 2n-gons.

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; updated 1 Sept 2020 with WebSketchpad.