## Related Variations & Generalizations of Napoleon's Theorem

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).

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } 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 ΔABC, so that ∠GAB = ∠HCB = α, ∠HBC = ∠IAC = β and ∠ICA = ∠GBA = γ and α + β + γ = 90o, 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 = 360o, 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).

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