## 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 *T*_{1} and *T*_{3} are the respective centroids of *ABE* and *CDG*, and *G*_{2} is the centroid of *EGF*, then ∠*T*_{1}G_{2}T_{3} = 120°, and *T*_{1}G_{2} = *G*_{2}T_{3}.

Corollary: If *G*_{4} is the centroid of *EGH*, then *T*_{1}G_{2}T_{3}G_{4} 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 Δ*BDF* 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.*

*Back
to "Dynamic Geometry Sketches"*

*Back
to "Student Explorations"*

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