Related Triangle Variations & Generalizations of Napoleon's Theorem

Napoleon Related triangle Variation 1
If similar triangles ABD, EBC and AFC are erected on the sides of any triangle ABC, and G, H and I are incentres of these triangles, then they form a triangle so that ∠G = ½(∠DAB + ∠DBA), ∠H = ½(∠EBC + ∠ECB) and ∠I = ½(∠FCA + ∠FAC) - De Villiers & Meyer (1995).

Napoleon Related Triangle Variations

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Napoleon Related Variation 2
If triangles AGB, HCB and IAC are erected on the sides of any ΔABC, so that ∠GAB = ∠HCB = α, ∠HBC = ∠IAC = β and ∠ICA = ∠GBA = γ and α + β + γ = 90^o, then ∠HGI = 2β, ∠GHI = 2γ and ∠GIH = 2α - De Villiers & Meyer (1995).
(To view & manipulate the dynamic version of this 2nd variation, navigate to it by clicking the 'Link to Napoleon related 2' button in the above dynamic sketch; the picture below is static).

Napoleon Related Variation 2 (1995)

Notes
1) If in the hexagon AGBHCI in the Related Variation 2 above, AG = AI, BG = BH and CH = CI (or GA = GB, HB = HC and IC = IA), we obtain a configuration meeting the conditions of Pompe's Hexagon Theorem (2016).
2) This variation can also be used to solve/prove this interesting Dirk Laurie Tribute Problem.

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Special Case of Napoleon Related Triangle Variation 2
If similar, acute-angled triangles DBA, CBE and CFA are erected on the sides of any triangle ABC, then the orthocentres of the three triangles form a triangle with ∠HGI = 2∠HBC, ∠GHI = 2∠ICA and ∠GIH = 2∠GAB - De Villiers (1996).
(To view & manipulate the dynamic version of the special case of this 2nd variation, navigate to it by clicking the 'Link to Napoleon special case' button in the dynamic sketch at the top; the picture below is static).

Special Case of Napoleon Related Variation 2

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Napoleon Related Variation 3 - Michael Fox's Theorem
If similar triangles DBA, CBE and CFA are erected on the sides of any triangle ABC, and points G, H and I are in the same relative positions in these triangles, then ∠HGI = ∠HBC + ∠IAC, ∠GHI = ∠ICA + ∠GBA, and ∠GIH = ∠GAB + ∠HCB - Fox (1998).
(To view & manipulate the dynamic version of this 3rd variation, navigate to it by clicking the 'Link to Michael Fox's Theorem' button in the dynamic sketch at the top; the picture below is static).

Napoleon Related Variation 3 - Michael Fox's Theorem (1998)

References
De Villiers, M. & Meyer, J. (1995). A generalized dual of Napoleon's theorem and some further extensions. Int. J. Math. Ed. Sci. Technol., 26(2), pp. 233-241.
De Villiers, M. (1994, 1996, 2009). Some Adventures in Euclidean Geometry (free to download). Lulu Press: Dynamic Mathematics Learning, pp. 179-180.
Egamberganov, K. (2017). A generalization of the Napoleon's Theorem. Mathematical Reflections, no. 3, pp. 1-7.
Fox, M. (1998). Napoleon triangles and adventitious angles. The Mathematical Gazette, Nov., pp. 413-415.

Egamberganov's Theorem
In a 2017 paper A generalization of the Napoleon's Theorem in Mathematical Reflections, no. 3, Khakimboy Egamberganov generalizes Napoleon's Theorem even further, and applies it to solve several interesting Olympiad type problems, including Pompe's Hexagon Theorem. The beauty of Egamberganov's Theorem is that it's also a generalization of all four the results above. His theorem can be seen as a hexagon theorem and can be formulated as follows in relation to the four sketches above:
Given a hexagon AGBHCI with AG* BH * CI = GB * HC * IA, and ∠AGB + ∠BHC + ∠CIA = 360^o, then ∠HGI = ∠HBC + ∠IAC, ∠GHI = ∠ICA + ∠GBA, and ∠GIH = ∠GAB + ∠HCB.
(A dynamic geometry sketch of this theorem will be done in due course & posted in a future update).

Related Links
Napoleon's Theorem (Rethinking Proof activity)
The Fermat-Torricelli Point (Rethinking Proof activity)
Some Circle Concurrency Theorems (Approaching Napoleon differently)
Napoleon's Theorem: Generalizations, Variations & Converses
Napoleon's Regular Hexagon
Some Hexagon Generalizations of Napoleon's Theorem
Pompe's Hexagon Theorem (Provides a direct proof of Napoleon's theorem)
Miquel's Theorem (Rethinking Proof activity)
A variation of Miquel's theorem and its generalization
Fermat-Torricelli Point Generalizations
Some Variations of Vecten configurations
Dirk Laurie Tribute Problem
The 120^o Rhombus (or Conjoined Equilateral Triangles) Theorem
Another concurrency related to the Fermat point of a triangle plus related results
Anghel's Hexagon Concurrency theorem

External Links
Napoleon's theorem (Wikipedia)
Napoleon's Theorem, Two Simple Proofs (Cut The Knot)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Free Download of Geometer's Sketchpad & Learning/Instructional Modules on various topics

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