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 *A _{1}*,

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 *A _{3}*,

Dao Than Oai’s generalization of Napoleon’s theorem

**Challenge**

Can you explain why (prove that) ∆*A _{1}B_{1}C_{1}*, ∆

**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 A _{2}B_{2}C_{2}**' button (where

1c) Given your observation in 1b) what do you suspect about ∆

1d) Check your conjecture in 1c) above, by clicking on the '

**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 ∆*A _{2}B_{2}C_{2}* and ∆

2b) Click on the '

**Further Generalization: Similar n-gons**

3a) Can you generalize further to similar quadrilaterals instead of similar triangles? Click on the '

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.

**Challenge**

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.*

**Another Proof**

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).

**Reference**

O.T. Dao. (2015). Two generalizations of the Napoleon theorem.

**Related Links**

Napoleon's 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

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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.