Given a tangential/circumscribed hexagon ABCDEF as shown below, what do you notice about the two sums of alternate sides?
Drag any of the tangent points to explore your observation.
Circumscribed Hexagon Alternate Sides Sum Theorem
Historical Note: This tangential/circumcribed hexagon theorem is a generalization of Pitot's Theorem for a tangential/circumscribed quadrilateral.
Challenge
1) Can you explain why (prove that) the tangential/circumscribed hexagon result is true?
2) If not, click on the given HINT button in the sketch.
3) Is the converse true? I.e. if AB + CD + EF = BC + DE + FA does it imply that ABCDEF is tangential/circumscribed? Investigate & prove or disprove.
4) Can you generalize further to tangential/circumscribed octagons, etc.? Explore dynamically!
5) Can you formulate a similar result for a cyclic hexagon involving its angles? Investigate further!
6) Regarding 4) & 5), go here for more information: Further Generalization & Dual.
Also see this related result for a tangential/circumscribed hexagon Triangulated tangential hexagon with incircles
a) A classroom activity and proof of the above tangential/circumscribed hexagon result is given in my Rethinking Proof with Sketchpad book which is now free to download (click on link).
b) Proofs and generalizations of these results are also given in Some Adventures in Euclidean Geometry, as well as generalizations to tangential/circumscribed 2n-gons with crossed sides. My book is now available to download for free at Some Adventures in Euclidean Geometry.
Related Links
Side Divider Theorem for a Circumscribed Quadrilateral
The Tangential (or Circumscribed) Polygon Side Sum theorem
Cyclic Hexagon Alternate Angles Sum Theorem
A generalization of the Cyclic Quadrilateral Angle Sum theorem
Angle Divider Theorem for a Cyclic Quadrilateral
More Area, Perimeter and Other Properties of Circumscribed Isosceles Trapeziums and Cyclic Kites (PDF)
Some Trapezoid (Trapezium) Explorations
Midpoint trapezium (trapezoid) theorem generalized
Tiling with a Trilateral Trapezium and Penrose Tiles (PDF)
Matric Exam Geometry Problem - 1949
Finding the Area of a Crossed Quadrilateral
The Parallel-pentagon and the Golden Ratio
International Mathematical Talent Search (IMTS) Problem Generalized
Clough's Theorem (a variation of Viviani) and some Generalizations
A Geometric Paradox Explained
Semi-regular Angle-gons and Side-gons: Generalizations of rectangles and rhombi
Intersecting Circles Investigation
SA Mathematics Olympiad Problem 2016, Round 1, Question 20
SA Mathematics Olympiad 2022, Round 2, Q25
An extension of the IMO 2014 Problem 4
A 1999 British Mathematics Olympiad Problem and its dual
Dirk Laurie Tribute Problem
Golden Quadrilaterals
Crossed Quadrilateral Properties
Extangential Quadrilateral
Triangulated Tangential Hexagon theorem
Theorem of Gusić & Mladinić
Conway's Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Pirate Treasure Hunt and a Generalization
A Quarter Circle Investigation, Explanation & Generalization
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Modified by Michael de Villiers, 24 March 2012; updated 1 Sept 2020 with WebSketchpad; updated 13 March 2025.