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.

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 Alternate Angles Sum Cyclic Hexagon theorem and Alternate Sides Sum Circumscribed Hexagon theorem.

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 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.


Back to "Dynamic Geometry Sketches"

Back to "Student Explorations"

Created by Michael de Villiers, 27 March 2012; updated 1 Sept 2020 with WebSketchpad.