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.

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