**Napoleon Related 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*).

Napoleon Related Variation 1 (1995)

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**Napoleon Related Variation 2**: If triangles *AGB*, *HCB* and *IAC* are erected on the sides of any Δ*ABC*, 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.

**Important**: To view & manipulate the *dynamic version* of this 2nd generalization, navigate to it using the appropriate button in the **ABOVE** dynamic sketch; the picture below is static.

Napoleon Related Variation 2 (1995)

**Note**:

1) If in the hexagon *AGBHCI* 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) It can also be used to solve/prove this interesting Dirk Laurie Tribute Problem.

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**Special Case of Napoleon Related 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*.

**Important**: To view & manipulate the *dynamic version* of this 2nd generalization, navigate to it using the appropriate button in the dynamic sketch right 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*.

**Important**: To view & manipulate the *dynamic version* of this 2nd generalization, navigate to it using the appropriate button in the dynamic sketch right at the **TOP**; the picture below is static.

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

**References**

i) 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.

ii) De Villiers, M. (1996). *Some Adventures in Euclidean Geometry*, pp. 179-180.

iii) Fox, M. (1998). Napoleon triangles and adventitious angles. *The Mathematical Gazette*, Nov., pp. 413-415.

**Egamberganov's Theorem**: In a 2017 paper in *Mathematical Reflections*, no. 3, Khakimboy Egamberganov generalizes Napoleon's Theorem even further A generalization of Napoleon's Theorem, 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 the next update).

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Created in 2008 by Michael de Villiers, modified 5 April 2020.