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 A_{1}A_{2}A_{3}A_{4}A_{5}A_{6} 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 G_{1}, G_{3} & G_{5} are the respective centroids of B_{6}B_{1}B_{2}, B_{2}B_{3}B_{4} and B_{4}B_{5}B_{6}, then G_{1}G_{3}G_{5} form an equilateral triangle. (Similarly, the respective centroids of B_{1}B_{2}B_{3}, B_{3}B_{4}B_{5} and B_{5}B_{6}B_{1}, form an equilateral triangle G_{2}G_{4}G_{6}).

If, for example, we let points A_{1} and A_{6} coincide, as well as A_{2} and A_{3}, and A_{4} and A_{5}, 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) △G_{1}G_{3}G_{5} 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 G_{2}G_{4}G_{6}.
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.