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.

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This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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Let A_m, B_m, C_m be the midpoints of  B''C'', A''C'', and A''B'', respectively. Then triangles AB_mC_m, A_mBC_m, A_mB_mC 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, λ² - λ + 1 = 0.

We have

	α''	= γ + λ(β - γ),
	β''	= α + λ(γ - α),
	γ''	= β + λ(α - β).

Further

	αm	= (β'' + γ'')/2	= (α + β)/2 + λ(γ - β)/2,
	βm	= (α'' + γ'')/2	= (β + γ)/2 + λ(α - γ)/2,
	γm	= (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 α = A, where A = β_m + λ(γ_m - β_m).

Well, γ_m - β_m = (γ - β)/2 + λ(2α - β - γ)/2. Thus

	A	= βm + λ(γm - βm)
		= (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_mC_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

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

Napoleon's Theorem

• Napoleon's Theorem
• A proof with complex numbers
• A second proof with complex numbers
• A third proof with complex numbers
• Napoleon's Theorem, Two Simple Proofs
• Napoleon's Theorem via Inscribed Angles
• A Generalization
• Douglas' Generalization
• Napoleon's Propeller
• Napoleon's Theorem by Plane Tessellation
• Fermat's point
• Kiepert's theorem
• Lean Napoleon's Triangles
• Napoleon's Theorem by Transformation
• Napoleon's Theorem via Two Rotations
• Napoleon on Hinges
• Napoleon on Hinges in GeoGebra
• Napoleon's Relatives
• Napoleon-Barlotti Theorem
• Some Properties of Napoleon's Configuration
• Fermat Points and Concurrent Euler Lines I
• Fermat Points and Concurrent Euler Lines II
• Escher's Theorem
• Circle Chains on Napoleon Triangles
• Napoleon's Theorem by Vectors and Trigonometry
• An Extra Triple of Equilateral Triangles for Napoleon
• Joined Common Chords of Napoleon's Circumcircles
• Napoleon's Hexagon
• Fermat's Hexagon
• Lighthouse at Fermat Points
• Midpoint Reciprocity in Napoleon's Configuration
• Another Equilateral Triangle in Napoleon's Configuration
• Yet Another Analytic Proof of Napoleon's Theorem
• Leo Giugiuc's Proof of Napoleon's Theorem
• Gregoie Nicollier's Proof of Napoleon's Theorem
• Fermat Point Several Times Over

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