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 Savarn’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?
(If stuck, a proof is given in the reference below).

Reference
Humenberger, H.; Schuppar, B. & De Villiers, M. (2022). Jha and Savarn’s generalisation of Napoleon’s theorem. Global Journal of Advanced Research on Classical and Modern Geometry (GJARCMG), Vol.11, Issue 2, pp. 190-197.

Related Links
Jha and Savarn’s generalisation of Napoleon’s theorem
Pompe's Hexagon Theorem
The Center of Gravity (Centroid) of a Triangle (Rethinking Proof activity)
The Fermat-Torricelli Point (Rethinking Proof activity)
Napoleon's Theorem (Rethinking Proof activity)
Miquel's Theorem (Rethinking Proof activity)
Napoleon's Theorem: Generalizations, Variations & Converses
Napoleon's Regular Hexagon
Triangle Centroids of a Hexagon form a Parallelo-Hexagon: A generalization of Varignon's Theorem
Point Mass (Vertex) Centroid (centre of gravity or balancing point) of Quadrilateral
Centroid of Cardboard (Lamina) Quadrilateral
Centroid (balancing point) of Perimeter Quadrilateral
Van Aubel's Theorem and some Generalizations
Van Aubel Vertex Centroid & its Generalization
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
Geometry Loci Doodling of the Centroids of a Cyclic Quadrilateral

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