Additional properties of a particular hexagon

Note: The properties of the particular hexagon below are additional to those given at Concurrency, collinearity and other properties of a particular hexagon.

Theorem 1
Given a hexagon ABCDEF with AB = BC, CD = DE, EF = FA, and ∠A = ∠C = ∠E, then ABCDEF has an incircle.
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 ABCDEF with AB = BC, CD = DE, EF = FA, and ∠A = ∠C = ∠E, then the angle bisectors of ∠A, ∠C, and ∠E are concurrent at the circumcentre, Q, of △ACE.
Important Note: The concurrency of the main diagonals of the hexagon ABCDEF given above now follows directly from Theorem 1 with the application of Brianchon's theorem, and therefore provides a much easier proof than the one given by us in our paper in the link at the top (which uses a theorem by Anghel (2016)).
Acknowledgement: It has also come to our attention that this hexagon concurrency theorem is apparently attributed to A. Zaslavsky, and a diagram (without proof) of it is given in Arseniy Akopyan's (2011) diagrammatic book Geometry in Figures (problem 4.9.26, p. 53). It also appeared earlier as a problem in the Third Sharygin Olympiad in Geometry, 2007, Final Round, Grade 9, Problem 3. Though in Russian, it's easy to see that the given solution on p. 6 to Problem 3 of the Sharygin Solutions, is via Theorem 1 above (see p. 6, Fig. 9.3).

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). Further reflections on a particular hexagon. 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
Concurrency, collinearity and other properties of a particular hexagon.

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

Additional properties of a particular hexagon

.sketch_canvas { border: medium solid lightgray; display: inline-block; } Additional properties of a particular hexagon

Additional properties of a particular hexagon