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.

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 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 tangential/circumscribed 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.

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Modified by Michael de Villiers, 24 March 2012; updated 1 Sept 2020 with WebSketchpad.