## 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 Bi 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.

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } 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.

Created 16 June 2022 by Michael de Villiers, using WebSketchpad.; updated 15 August 2022; 2 October 2022; 6 July 2023; 20 Nov 2023