Given a cyclic hexagon ABCDEF as shown below, what do you notice about the two sums of alternate angles?
Drag any of the vertices to explore your observation.
Alternate Angles Sum Cyclic Hexagon Theorem
Historical Note: This cyclic hexagon theorem does not appear in Euclid's "Elements", and was apparently first discovered and proved by Duncan Gregory who in 1836 published it in the Cambridge Mathematical Journal.
1) Can you explain why (prove that) the cyclic hexagon result is true?
2) If not, click on the given HINT button in the sketch.
3) Can you prove the result in more than one way? Check your solutions against those in this 2017 Multiple Solutions paper by Duncan Samson who tried this problem with his high school class at St. Andrews & DSG.
4) Is the converse true? I.e. if ∠A + ∠C + ∠E = ∠B + ∠D + ∠F does it imply that ABCDEF is cyclic? Investigate & prove or disprove.
5) Check your answer in regard to 4) by reading this paper Recycling cyclic polygons dynamically.
6) Can you generalize further to cyclic octagons, etc.? Explore dynamically!
7) Can you formulate a similar result for a tangential/circumscribed hexagon involving its sides? Investigate further!
8) Regarding 6) & 7), go here for more information: Further Generalization & Dual.
a) A classroom activity and proof of this cyclic hexagon result is also given in Rethinking Proof with Sketchpad.
b) Proofs and generalizations of these results are also given in Some Adventures in Euclidean Geometry, as well as generalizations to cyclic 2n-gons with crossed sides. The book 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.
Modified by Michael de Villiers, 24 March 2012; updated 1 Sept 2020 with WebSketchpad.