**Investigate & Conjecture**

Given a (convex) 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.

Cyclic Hexagon Alternate Angles Sum Theorem

**Historical Note**: This (convex) 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*.

**Cyclic Quadrilateral Note**: This result is a generalization to cyclic hexagons of the familiar result for a cyclic quadrilateral *ABCD*, namely, ∠*A* + ∠*C* = ∠*B* + ∠*D* (which is equal to 180^{o} when it is convex, and using directed angles, equal to 360^{o} when crossed (see De Villers, 1994, 1999)^{1}). While it is usually formulated at high school level for convex cyclic quadrilaterals that the sum of *opposite* angles are supplementary, it is actually more convenient to regard the angles involved as 'alternate' angles, which more clearly illustrates the hexagon generalization above, and to cyclic 2*n*-gons in general.

**Challenge**

1) Can you explain why (prove that) the sum of the *alternate* angles of a cyclic hexagon remain equal, specifically that ∠*A* + ∠*C* + ∠*E* = ∠*B* + ∠*D* + ∠*F*?

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. More recently in 2023, Yiu-Kwong Man provided a nice Visual Proof of the convex case.)

**Converse**

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.

**Further Generalization**

6) Can you generalize further to (convex) cyclic octagons, cyclic decagons, etc.? Explore dynamically on your own using appropriate software!

**Dual Result**

7) Can you formulate a similar dual result for a tangential/circumscribed hexagon involving its *sides*? Investigate further!

**Related Links**

8) Regarding 6) & 7) above, go here for more information: Further Generalization & Dual and Circumscribed Hexagon Alternate Sides Sum theorem respectively.

**Crossed Cyclic Hexagon Investigation**

9) What happens to the sum of the alternate angles when the hexagon *ABCDEF* becomes crossed?

10) To investigate this question click on the '**Link to crossed cyclic hexagon - directed angles**' button to navigate to a new sketch. This sketch uses 'directed angles' which means that according to the same convention used in trigonometry, counter-clock wise rotations (angles) are regarded as positive and counter-clock wise ones as negative.

**Investigate**: Drag any one of the vertices past an adjacent one. What do you notice about the sum of the alternate angles now? Ensure that you at least drag your sketch into each of the three configurations below.

11) **Challenge**: Can you explain why (prove that) your observations in 10) above are true?

12) **Check**: Click on the 'Link to Check Findings' button to check your findings in 10) & 11) above.

**Footnote to Cyclic Quadrilateral Note**

^{1}If angles for a crossed cyclic quadrilateral are measured in the same way as with the crossed hexagon above, using directed angles, and not allowing any reflexive angles, but measuring negative angles instead, then the sum of alternate angles for a crossed cyclic quadrilateral would be 0^{o}. This shows how the choice of one's definitions in mathematics can profoundly influence one's results.

**Published Paper**

A paper Investigating Alternate Angle Sums of Crossed Cyclic Hexagons of mine related to the above has been published in the *Learning & Teaching Mathematics* journal, No. 35, Dec 2023, pp. 28-31.

...........

**Using GeoGebra to investigate alternate angle sums in crossed cyclic hexagons **

GeoGebra Classic 6 automatically measures angles using directed angles in the background and can also measure *reflexive* angles (i.e. greater than 180^{o}). See the PDF Cyclic Hexagon Illustrations with GeoGebra to see some examples of how it is different from the Sketchpad sketches above.

As can be seen in the preceding PDF link, for crossed cyclic hexagons, the sum of the alternate angles with *GeoGebra* can be 540^{o}, 720^{o} and 900^{o}; so there are also three different cases like with Sketchpad above (where the three measured sums are -180^{o}, 0^{o} and 180^{o}). The difference is due to *GeoGebra* measuring directed angles greater than 180^{o} whereas *Sketchpad* not only measures, but also displays negative angles, and is restricted to only measuring directed angles smaller than 180^{o}.

However, since *GeoGebra* takes the absolute values of the directed angles, it does not display negative angles, and in some configurations, **the two sums of alternate angles are unfortunately not displayed as being equal**. One example of this issue is given in the PDF at the preceding link where the sum of one set of alternate angles is 540^{o} while the other is given as 180^{o}.

**Additional References**

a) A classroom activity and guided proof of the (convex) cyclic hexagon result is also given in the FREE DOWNLOAD of Rethinking Proof with Sketchpad (1999, 2003, 2012), pp. 50; 162-163.

b) Proofs and generalizations of these results are also given in *Some Adventures in Euclidean Geometry (1994)*, as well as generalizations to cyclic 2*n*-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.

c) De Villiers, M. (2020). The Value of using Signed Quantities in Geometry *Learning & Teaching Mathematics*, no. 29.

**Related Links**

Crossed Quadrilateral Properties

Interior angle sum of polygons (incl. crossed): a general formula

Investigating a general formula for the interior angle sum of polygons (A suggested guided learning activity starting with LOGO (turtle) geometry)

A generalization of the Cyclic Quadrilateral Angle Sum theorem

Alternate Sides Sum Circumscribed Hexagon

Parallelogram Distances

2D Generalizations of Viviani's Theorem

Angle Divider Theorem for a Cyclic Quadrilateral

Side Divider Theorem for a Circumscribed/Tangential Quadrilateral

Conway’s Circle Theorem as special case of Side Divider Theorem

Pitot's Theorem (alternate sides sum theorem for a tangential quadrilateral)

Tangential Quadrilateral Theorem of Gusić & Mladinić

Semi-regular Angle-gons and Side-gons: Generalizations of rectangles and rhombi

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Created in 2008; modified by Michael de Villiers, 24 March 2012; updated 1 Sept 2020 with *WebSketchpad*; updated 27 June 2023; 4 July 2023; 12 Dec 2023.