Napoleon's Theorem

The dynamic geometry activities below are from the "Proof as Challenge" section of my book Rethinking Proof (free to download).

Worksheet & Teacher Notes
Open (and/or download) a guided worksheet and teacher notes to use together with the dynamic sketch below at: Napoleon Worksheet & Teacher Notes.

Prerequisites
Students should know that the opposite (alternate) angles in a (convex) quadrilateral are supplementary, if and only if, the (convex) quadrilateral is cyclic. Students should be familiar with the definition of a kite as a quadrilateral with two (different) pairs of adjacent sides equal, and the property that its diagonals are perpendicular. Specifically, it is highly recommended that students have already completed the following learning activities:
Exploring Kite Properties (Suggested Grades 4 - 9)
Cyclic Quadrilateral (Opposite ∠'s supplementary)
Cyclic Quadrilateral Converse (Converse of result above)
The Fermat-Torricelli Point of a Triangle

The dynamic sketch below shows a triangle ABC with equilateral triangles on the sides as well as that the three circumcircles of the equilateral triangles are concurrent. This circle concurrency was discovered and proved in a previous activity: The Fermat-Torricelli Point of a Triangle. In the guided activity below you will discover and prove a beautiful property about the triangle formed by the circumcentres of these equilateral triangles.

Napoleon's Theorem

Historical Background
While this beautiful theorem is named after the famous French Emperor, Napoleon Bonaparte (1769-1821), and that he apparently was interested in geometry, it is doubtful as pointed out by Grünbaum (2012) that he actually discovered and proved this theorem himself. The earliest known published formulation of the result was given by W. Rutherford, in 1825 in The Ladies’ Diary. No mention is made of Napoleon in that paper and even though the result is closely connected to the Fermat-Torricelli point, and it may be that Torricelli or his student Viviani might have discovered it, there is no evidence of that in their published papers. Apparently the earliest direct attribution of the result to Napoleon was in 1911, in the publication of an Italian geometry text book by the mathematician, Aureliano Faifofer.

Note
1) Users are strongly encouraged to download the accompanying Worksheet, and to print it out to use in conjunction with the dynamic sketch above. Alternatively, copy the URL: https://dynamicmathematicslearning.com/napoleon-rethinking-proof.pdf and paste it into a new browser window. Then resize the new window to place it side by side with this one, or one below the other. (However, this is not likely to be a feasible option for users using small screens such as a cellphone or tablet.)

Explore
2) Use the 'Segment' Tool in the Toolbar on the left to construct △GHI.
3a) What type of triangle does GHI look like? Check your intuitive guess by dragging.
Also measure the lengths of the sides by the 'Distance' Tool in the Toolbar on the left (scroll down if necessary).
3b) (Alternatively to 3a), click on the 'Show side lengths' button).
4) What do you notice in 3) above? Can you formulate a conjecture?
5) If you drag ABC into a degenerate triangle so that A, B and C lie in a straight line, does the result still hold?
6) Drag ABC until the equilateral triangles all lie towards the inside. Does the result still hold?
7) Formulate your conjecture in words.

Challenge
8) Can you prove your conjecture in 7) above? Can you prove it in more than one way?

Proving
9) Work through the questions in the guided proof in the accompanying Worksheet, using the four top buttons on the right as indicated in the text (the 'Show segments' button as well as the three 'Show quadrilateral' buttons). The proof requires only knowledge of the properties of cyclic quadrilaterals and kites.

Further Generalizations
Since equilateral triangles are all similar to each other, it is natural to wonder and further explore what happens if similar triangles are constructed on the sides of a triangle. (Also compare with the Fermat-Torricelli activity).
Here are four possible further generalizations of Napoleon's theorem:
10) If similar triangles DBA, BEC, and ACF are erected on the sides of any triangle ABC, their circumcenters G, H, and I form a triangle similar to the three triangles.
11) If similar triangles DBA, CBE, and CFA are erected on the sides of any triangle ABC, their circumcenters G, H, and I form a triangle similar to the three triangles.
(Notice that the similar triangles have different orientations in the two different generalizations in 10) & 11).)
12) If triangles DBA, BEC, and ACF (or DBA, CBE, and CFA) are erected on the sides of any triangle ABC so that ∠D + ∠E + ∠F = 180°, then their circumcircles meet in a common point (are concurrent), and their circumcenters G, H, and I form a triangle, then ∠G = ∠D, ∠H = ∠E, and ∠I = ∠F.
(Notice that the result in 12) generalizes both 10) and 11)).
13) Click on the 'Link to similar △'s' button to navigate to a new dynamic sketch which illustrates the 4th generalization:
If similar triangles DBA, BEC, and ACF are erected on the sides of any triangle ABC, and any three points P, Q, and R are chosen so that they respectively lie in the same positions relative to these triangles, then P, Q, and R form a triangle similar to the three triangles.
(Notice that the result in 13) further generalizes 10).
Challenge
14) Can you prove the generalization given in 12)?
(Compare your proof with one for generalization 12) given in the accompanying Teacher Notes, pp. 195-196).
15) Can you prove the generalization given in 13)?
(Proofs for 13) can be found in De Villiers (1994, 2009: 177–181), De Villiers & Meyer (1995) and King (1997).
Explore More
16) Explore some more triangle variations, and generalizations to hexagons, respectively at:
Related Triangle Variations & Generalizations
Some Hexagon Generalizations

Note: Dynamic geometry sketches for the above generalizations are also available at: Triangle Generalizations of Napoleon's Theorem.

Comment: Mathematicians have been fascinated by Napoleon's theorem for the past two centuries and so many papers have been around written about it that it is impossible to reference them all here. So the list of references given below is merely a sample of some of the available literature on the topic.

References
Boutte, G. (2002). The Napoleon Configuration. Forum Geometricorum, Vol 2, 39–46.
Coxeter, H. & Greitzer, S. (1967). Geometry Revisited, pp. 61-65. Washington: DC, MAA.
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.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2022). Some Circle Concurrency Theorems. Learning and Teaching Mathematics, No. 33 Dec), pp. 34-38.
De Villiers, M. (2024). A Surprise Equilateral Triangle. Learning and Teaching Mathematics, No. 37, pp. 27-29.
Grünbaum, B. (2012). Is Napoleon’s Theorem Really Napoleon’s Theorem? American Mathematical Monthly, 119, June–July, 495-501.
Hungerbühler, N. (2025). When Varignon meets Napoleon. Journal of Geometry, 116(2), 1-14. DOI: https://doi.org/10.1007/s00022-025-00746-9
King, J. 1997. An eye for similarity transformations. Geometry Turned On! Edited by D. Schattschneider & J. King. Washington, D.C.: Mathematical Association of America.
Smyth, M. R. F. (2007). MacCool’s Proof of Napoleon’s Theorem. Irish Math. Soc. Bulletin, 59, 71–77.
Shirali, S. (2016). Napoleon's Theorem, Part 1. At Right Angles, Vol. 5, No. 3 (Nov.), 5-11.
Shirali, S. (2017). Napoleon's Theorem, Part 2. At Right Angles, Vol. 6, No. 3 (Nov.), 28-30.

Other Rethinking Proof Activities
Other Rethinking Proof Activities

External Links
Napoleon's theorem (Wikipedia)
Napoleon's Theorem, Two Simple Proofs (Cut The Knot)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
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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Created by Michael de Villiers with WebSketchpad, 2 April 2026; updated 5 April 2026.