Haag Hexagon - Extra Properties

A Haag-hexagon can be defined as a hexagon ABCDEF with AB = AF, CB = CD, ED = EF, and ∠A = ∠C = ∠E = 120o. One way of constructing a Haag hexagon is to start with an equilateral triangle ACE as described at Haag Construction (see Schattschneider, 1990, p. 90; De Villiers, 2014). Apart from tiling the plane, it also has the interesting concurrency property stated in Theorem 1 below (which can be easily proved from Jacobi's Generalization of the Fermat-Torricelli point.)

Theorem 1
Given a Haag hexagon ABCDEF with AB = AF, CB = CD, ED = EF, and ∠A = ∠C = ∠E = 120o, then AD, BE, and CF are concurrent at X.

In addition, the Haag Hexagon has the following interesting properties:

Theorem 2
The point X defined as in Theorem 1, the powerpoint P of (equilateral) △ACE, and the circumcentre O of △BDF are collinear.

Theorem 3
The 7 points B, D, F, P, X, the orthocentre H1 of △BDF, and the centre H2 of equilateral △ACE lie on a rectangular hyperbola. Likewise, the 6 points A, C, E, X, O and the centre H2 of equilateral △ACE lie on a rectangular hyperbola.
In the sketch below, click on the Show Hyperbola buttons to view & interact with the sketch dynamically illustrating this theorem.

Haag Hexagon - Extra Properties

Challenge
Can you prove Theorems 2 and 3 above?

Note
Theorems 2 and 3 can be proved in a similar way, using the same theorems, as the ones used in a joint paper by myself and Tran Quang Hung, Vietnam, for analogous results for a different hexagon ABCDEF with AB = BC, CD = DE, EF = FA, but with ∠A=∠C =∠E - go HERE to compare with the theorems above. This joint paper of ours, Concurrency, collinearity and other properties of a particular hexagon, has been published in the Mathematics Competitions Journal, Vol 35, No 1, 2022, pp. 82-91 of the World Federation of National Mathematics Competitions (WFNMC). All rights reserved.

References
1) De Villiers, M.D. (2014). An Investigation of Some Properties of the General Haag Polygon. Mathematics in School, 43(3), 15-18.
2) Schattschneider, D. (1990). M.C. Escher: Visions of Symmetry. New York: W.H. Freeman & Co.

Related Links
Haag Hexagon and its generalization to a Haag Polygon
Concurrency, collinearity and other properties of a particular hexagon
Jacobi's Generalization of the Fermat-Torricelli point
Power Lines of a Triangle
Easy Hexagon Explorations


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Created by Michael de Villiers, 20 June 2022 with WebSketchpad; updated 2 Sept 2022; 6 March 2024.