## The 120o Rhombus (or Conjoined Twin Equilateral Triangles) Theorem

The 120o Rhombus (or 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. Further, if G4 is the centroid of EGH, then T1G2T3G4 is a rhombus (consisting of two equilateral triangles T1G2G4 and T3G2G4 joined along mutual side G2G4).

This theorem is particularly useful in proving Jha and Savaran’s generalisation of Napoleon’s theorem.

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } The 120o Rhombus (or Conjoined Twin Equilateral Triangles) Theorem

Challenge
Can you explain why (prove that) the above theorem is true?

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, using WebSketchpad; updated 15 August 2022.