**Note**: The properties of the particular hexagon below are additional to those given at

Given a hexagon

Use the sketch below to view & interact with a dynamic illustration of this theorem.

An equivalent formulation of Theorem 1 is the following: Given a hexagon

**Converse of Theorem 1**

The alternative formulation of theorem 1 now gives us the following neat converse: Given a hexagon *ABCDEF* with *AB* = *BC*, *CD* = *DE*, *EF* = *FA*, and the angle bisectors of ∠*A*, ∠*C*, and ∠*E* are concurrent at the circumcentre, *Q*, of △*ACE*, then ∠*A* = ∠*C* = ∠*E*.

**Theorem 2**

Given a hexagon *ABCDEF* with *AB* = *BC*, *CD* = *DE*, *EF* = *FA*, and ∠*A* = ∠*C* = ∠*E*, then *ABCDEF* is cyclic only when △*ACE* is equilateral, and the hexagon is regular.

Click on the *Link to Cyclic Exploration* button below to dynamically explore this theorem with an interactive sketch.

Additional properties of a particular hexagon

**Challenge**

Can you prove Theorem 1 and its converse, as well as Theorem 2?

**Further Generalization**

Can you generalize Theorem 1 and its converse to an octagon, decagon, etc.?

1) Click on the *Link to Octagon by reflection* to explore a generalization of the converse of Theorem 1 to an octagon,

2) Click on the *Link to Octagon Check* buttons to check whether Theorem 1 similarly generalizes to an octagon. What do you notice?

3) Click on the *Link to Decagon* to explore a generalization of both Theorem 1 and its converse to a decagon.

**Challenge**: Can you prove the generalizations above?

**Reference**

De Villiers, M. (2022). *Mathematics Competitions Journal*, Vol 35, No 2, 2022, pp. 44-51. Published by the World Federation of National Mathematics Competitions (WFNMC). *All rights reserved*.

**More Properties of this Particular Hexagon**

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Created by Michael de Villiers, 24 September 2022 with *WebSketchpad*, updated 26 September 2022, 12 March 2023.