Jha and Savaran’s generalisation of Napoleon’s theorem

In December 2021, two Grade 11 schoolboys from India, Jayendra Jha and Sankalp Savaran, using dynamic software discovered the following apparently new generalisation of Napoleon’s theorem, and posted the problem at Stack Exchange.

Theorem: Given a hexagon A₁A₂A₃A₄A₅A₆ with equilateral triangles constructed on the sides, either inwardly or outwardly, and the apexes of the equilateral triangles labelled B_i as shown below. If G₁, G₃ & G₅ are the respective centroids of B₆B₁B₂, B₂B₃B₄ and B₄B₅B₆, then G₁G₃G₅ form an equilateral triangle. (Similarly, the respective centroids of B₁B₂B₃, B₃B₄B₅ and B₅B₆B₁, form an equilateral triangle G₂G₄G₆).

If, for example, we let points A₁ and A₆ coincide, as well as A₂ and A₃, and A₄ and A₅, then Jha and Savaran’s result reduces to Napoleon’s theorem. The reader is invited to drag the dynamic figure below into this special case.

Jha and Savaran’s generalisation of Napoleon’s theorem

Challenge
Can you explain why (prove that) △G₁G₃G₅ is equilateral?
Here are Hints at proving the result.

Explore More
1) Click on the 'Link to both equilateral triangles' to navigate to a dynamic sketch which also shows the other equilateral triangle G₂G₄G₆.
2) What do you notice about the relationship between the two equilateral triangles?
3) Can you explain why (prove) this relationship between them holds? Here is a Hint to prove your observation about the relationship between the two equilateral triangles.

Note: Read a joint paper Jha and Savaran’s generalisation of Napoleon’s theorem by myself, together with Hans Humenberger, University of Vienna, and Berthold Schuppar, Technical University Dortmund, containing geometric proofs of the above results, in the Global Journal of Advanced Research on Classical and Modern Geometry (GJARCMG), Vol.11, (2022), Issue 2, pp. 190-197. All Rights Reserved.

Other Related Hexagon Generalizations of Napoleon's Theorem
O.T. Dao. (2015). Two generalizations of the Napoleon theorem.
Stachel, H. (2002). Napoleon’s Theorem and Generalizations Through Linear Maps, Contributions to Algebra and Geometry, Volume 43, No. 2, 433-444.

Related Links
Napoleon's Theorem
Some Generalizations of Napoleon's Theorem
Pompe's Hexagon Theorem
Related Variations & Generalizations of Napoleon's Theorem
Dao Than Oai’s hexagon generalization of Napoleon’s theorem
Some Napoleon Converses