## Hints for Proof of Jha and Savaran’s generalisation of Napoleon’s theorem

Jha and Savaran’s generalisation of Napoleon’s theorem can be very easily proved with the following two theorems:

Conjoined Twin Equilateral Triangles Theorem
Given a quadrilateral ABCD with three equilateral triangles ABE, BCF and CDG constructed on its sides, all inwardly or outwardly. If T1 and T3 are the respective centroids of ABE and CDG, and G2 is the centroid of EGF, then ∠T1G2T3 = 120°, and T1G2 = G2T3.
Corollary: If G4 is the centroid of EGH, then T1G2T3G4 is a rhombus.
To view & interact with a dynamic sketch illustrating this theorem, click Here.

Pompe’s Theorem
Given a hexagon ABCDEF with AB = BC, CD = DE and EF = FA, and angles ∠B + ∠D + ∠F = 360°, then the respective angles of ΔBDF are ∠B/2, ∠D/2 and ∠F/2.
To view & interact with a dynamic sketch illustrating this theorem, click Here.

Reference: 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.

Created 16 June 2022 by Michael de Villiers; updated 15 August 2022.