Dirk Laurie Tribute Problem

This nice problem is dedicated to Dirk Laurie (1946-2019), who gave it to me around 2010. He was an outstanding mathematician, and colleague for almost 40 years, and passed away suddenly in 2019. He and I worked together on the SA Mathematics Olympiad Committee for many years. He was also the South African IMO team leader on several occasions as well as the Chief Coordinator at IMO 2014 in Cape Town. Read a short tribute to him at: "Passing of a remarkable South African mathematician" or this scanned report (in Afrikaans) in the 18 August 2019 Rapport newspaper at: "Briljantste' mens skryf oor 'Onvolkome Kennis'".

Problem
Given a cyclic hexagon ABCDEF with alternates sides AB = CD = EF = R (the radius of the circle)¹, prove that the respective midpoints G, H and I of the other set of alternate sides FA, BC and DE form an equilateral triangle.

 

Dirk Laurie Tribute Problem

Challenge
1) Can you explain why (prove that) the above result is true?
2) Though the problem can probably be most easily attacked using trigonometry, the real challenge is finding a purely geometric solution. Can you do so?
3) If you get stuck, have a look at Pompe's Hexagon Theorem. Or alternatively, try using Napoleon Related Variation 2 on this page Related Variations & Generalizations of Napoleon's Theorem.
4) Can you generalize the result? Hint: is it necessary that the outer six vertices of the equilateral triangles that form the hexagon are cyclic?
5) Check your exploration in 4 above at: Some further generalizations.

Some worked solutions
1) Read the following solutions (2020) by Michael Fried (Israel) to Dirk Laurie's Hexagon Problem.
2) Here are some more solutions by Kousik Sett (India) and others published in the online journal Mathematical Reflections, no. 3, 2025 at: Solutions from Mathematical Reflections.

Footnote 1: In general, a cyclic hexagon with a set of alternate sides equal has 3 pairs of adjacent angles equal and can be viewed as one possible generalization of an isosceles trapezium. More information is available at Alternate sides cyclic-2n-gons.

Some Related Links
Dirk Laurie Tribute Problem Generalizations
Napoleon's Theorem
Napoleon's Theorem (a Rethinking Proof learning activity)
Airport Problem (a Rethinking Proof learning activity)
Alternate sides cyclic-2n-gons: generalization of isosceles trapezia
Some Triangle Generalizations of Napoleon's Theorem
Weighted Airport Problem
Related Variations & Generalizations of Napoleon's Theorem
Jha and Savarn’s generalisation of Napoleon’s theorem
The 120^o Rhombus (or Conjoined Twin Equilateral Triangles) Theorem
Dao Than Oai’s generalization of Napoleon’s theorem
Some Napoleon Converses
Pompe's Hexagon Theorem
A variation of Miquel's theorem and its generalization
Some Variations of Vecten configurations
Some Circle Concurrency Theorems
Attached Regular Pentagons form Congruent Equilateral Triangles
Semi-regular Angle-gons and Side-gons: Generalizations of rectangles and rhombi
An extension of the IMO 2014 Problem 4

Some External Links
Asymmetric Propeller
A Case of Similarity
A "propeller tétel" - és története

********************************