Dao Than Oai (2015) from Vietnam discovered the following interesting generalisation of Napoleon’s theorem.
Dao Than Oai's Theorem
Given a hexagon ABCDEF with equilateral ∆'s ABG, DHC, IEF constructed on the alternate sides AB, CD and EF, either inwardly or outwardly. Let A1, B1, C1 be the centroids of ∆FGC, ∆BHE, and ∆DIA respectively, let A2, B2, C2 be the centroids of ∆DGE, ∆AHF, and ∆BIC respectively. Then ∆A1B1C1 and ∆A2B2C2 are equilateral triangles.
If, for example, we let points A and F coincide, as well as B and C, and D and E, then Dao Than Oai’s result reduces to Napoleon’s theorem. The reader is invited to drag the dynamic figure below into this special case.
Another equilateral triangle
Another equilateral triangle is also embedded in the same configuration, but is not mentioned in Dao (2015). Let A3, B3, C3 be the centroids of ∆IGH, ∆ACE, and ∆DBF respectively. Then ∆A3B3C3 is also an equilateral triangle.
Dao Than Oai’s generalization of Napoleon’s theorem
Can you explain why (prove that) ∆A1B1C1, ∆A2B2C2 and ∆A3B3C3 are equilateral triangles?
Further Generalization: Similar Triangles
1a) Can you generalize further to similar triangles on the alternate sides of the hexagon instead of equilateral ones? Click on the 'Link to similar triangles' button on the bottom right to check your conjecture.
1b) Click on the 'Show triangle A2B2C2' button (where A2, B2, C2 are the respective centroids of ∆DGE, ∆AHF, and ∆BIC as defined in the equilateral case). What do you notice? Can you explain your observation?
1c) Given your observation in 1b) what do you suspect about ∆A3B3C3 (where A3, B3, C3 are the respective centroids of ∆IGH, ∆ACE, and ∆DBF as defined in the equilateral case)? Would it be similar to the three similar triangles or not? Why or why not?
1d) Check your conjecture in 1c) above, by clicking on the 'Show triangle A3B3C3' button. What do you notice? Can you explain your observation?
Other Arrangements of Similar Triangles
2a) Explore other possible arrangements of the three directly similar triangles. Can you find different arrangements in which the triangles ∆A2B2C2 and ∆A3B3C3 (with vertices defined as before) are similar to the original three similar triangles?
2b) Click on the 'Link to similar triangles arrangement 2' and 'Link to similar triangles arrangement 3' buttons on the bottom right to check your explorations in 2a). What do you notice?
Further Generalization: Similar n-gons
3a) Can you generalize further to similar quadrilaterals instead of similar triangles? Click on the 'Link to similar quadrilaterals' button on the bottom right to check your conjecture.
3b) On your own, explore other different arrangements of the four directly similar quadrilaterals that would also produce a 'centroid' quadrilateral similar to them? Hint: Consider the cyclic permutations of their labels.
4) Can you now generalize further to directly similar pentagons, directly similar hexagons, etc.?
5) On your own, using dynamic geometry software or paper & pencil, explore special cases of 3) and 4) above by looking at the cases for squares, regular pentagons, etc.
Can you explain why (prove that) the observed generalizations & variations in 1-4 above hold?
Hint: Try using the ideas & references at A Fundamental Theorem of Similarity or at Final Chapter of the Asymmetric Propeller Story at Cut the Knot website.
More on Napoleon's Theorem & other Generalizations
Click on Napoleon's Theorem: Generalizations & Converses to navigate to various dynamic sketches & some related papers.
Published Paper: A paper Geometric Proofs and Further Generalizations of Dao Than Oai’s Napoleon Hexagon Theorem "" by Hans Humenberger, University of Vienna, and Berthold Schuppar, Technical University Dortmund, and myself, containing geometric proofs of the above results, has been been published in Global Journal of Advanced Research on Classical and Modern Geometry (GJARCMG), Vol.12, (2023), Issue 1, pp. 158-168. All Rights Reserved.
After posting a note about Dao Than Oai's generalization and our proofs (see link above) in the Romantics of Geometry group (note 12273) on Facebook, Marian Cucoanes from Focsani, Romania, produced the following different proof (posted on 1 July 2023).
O.T. Dao. (2015). Two generalizations of the Napoleon theorem.
Some Generalizations of Napoleon's Theorem
Related Variations & Generalizations of Napoleon's Theorem
Jha and Savaran’s generalisation of Napoleon’s theorem
Some Napoleon Converses
Created 14 November 2022 by Michael de Villiers, using WebSketchpad; updated 19 & 21 November 2022; 1 & 7 December 2022; 13 April 2023; 6 July 2023.