# Napoleon's Relatives

*Napoleon's theorem* claims that the centers A', B', C', of the equilateral triangles A''BC, AB''C, ABC'', erected on the sides (either all inwardly or all outwardly) of a given triangle ABC form an equilateral triangle. There is a small family of triangles - related to Napoleon's - that were discovered relatively recently [Grünbaum].

The applet below serves to illustrate the construction.

What if applet does not run? |

Let A_{m}, B_{m}, C_{m} be the midpoints of B''C'', A''C'', and A''B'', respectively. Then triangles AB_{m}C_{m}, A_{m}BC_{m}, A_{m}B_{m}C are equilateral. If A*, B*, C* are their respective centers then triangle A*B*C* is also equilateral.

Complex numbers come in handy for proving this results. Let α, β, γ correspond to points A, B, C; α'', β'', γ'' to A'', B'', C''; α_{m}, β_{m}, γ_{m} to A_{m}, B_{m}, C_{m}. Leta λ be a rotation by 60°: λ³ = -1,

We have

α'' | = γ + λ(β - γ), | |

β'' | = α + λ(γ - α), | |

γ'' | = β + λ(α - β). |

Further

α_{m} | = (β'' + γ'')/2 | = (α + β)/2 + λ(γ - β)/2, | |

β_{m} | = (α'' + γ'')/2 | = (β + γ)/2 + λ(α - γ)/2, | |

γ_{m} | = (α'' + β'')/2 | = (γ + α)/2 + λ(β - α)/2. |

We wish to establish that &alpha, β_{m}, γ_{m} form an equilateral triangle. Suffice it to show that _{m} + λ(γ_{m} - β_{m}).

Well, γ_{m} - β_{m} = (γ - β)/2 + λ(2α - β - γ)/2. Thus

A | = β_{m} + λ(γ_{m} - β_{m}) | |

= (β + γ)/2 + λ(α - γ)/2 + λ[(γ - β)/2 + λ(2α - β - γ)/2] | ||

= [(β + γ)/2 + (2α - β - γ)/2] + λ[(α - γ)/2 + (γ - β)/2 + (β - α)/2] | ||

= α, |

as needed. Note that triangles ABC and AB_{m}C_{m} have the same orientation. A proof for the triangle A*B*C* is not very much different, however its orientation is with a natural consequence for the proof.

All the above assertions hold when triangles ABC'', AB''C, and A''BC are drawn with a reversed orientation. The triangle A*B*C* will be different, but the sum of the areas of the two (such triangles formed with different orientations) is one fourth the area of ΔABC.

### References

- B. Grünbaum,
__A RELATIVE OF "NAPOLEON'S THEOREM"__,*Geombinatorics*10(2001), 116 - 121 (available online)

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