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

The 120^o 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.

The 120^o 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.