Napoleon's Theorem: Third Proof with Complex Numbers

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. In the proof we are going to use complex numbers. The proof comes from [Bollobás, pp. 124-125] where the author makes an observation that after the slogan 'let's use vectors and complex numbers' no more thinking is needed. While this is true that one of algebra's purposes and uses is to mechanize solving problems, this is a third proof of Napoleon's theorem that makes use of complex numbers. So that, perhaps, some deliberation as to which road to choose might follow a conscious decision to base a proof on complex numbers.

The applet below serves to illustrate the proof.

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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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What if applet does not run?

Let a = CA', b = AB', c = BC', j = e^iπ/3, the counterclockwise rotation through 60° so that j³ = -1 and j = 1 + j². We have A'B = ja, B'C = jb, and C'A = jc, and conclude that

0 = a + ja + b + jb + c + jc = (1 + j)(a + b + c)

so that

a + b + c = 0.

For ΔA'B'C' to be equilateral suffice it to have, say, A'B' = jA'C'. Let's see that this is indeed so:

	jA'C' - A'B'	= j(A'B + BC') + (B'C + CA')
		= j(ja + c) + (jb + a)
		= (j² + 1)a + jc + jb
		= j(a + b + c)
		= 0

and we are done.

References

1. B. Bollobás, The Art of Mathematics: Coffee Time in Memphis, Cambridge University Press, 2006

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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