# Napoleon's Theorem,

A second proof with complex numbers

On each side of a triangle, erect an equilateral triangle, lying exterior to the original triangle. Then the segments connecting the centroids of the three equilateral triangles themselves form an equilateral triangle.

Let the original triangle be $ABC\;$ with equilateral triangle $ABC_{1},\;$ $CAB_{1},\;$ and $BCA_{1}\;$ built on its sides. Think of all the vertices involved as complex numbers. We shall apply a classical criterion to the three equilateral triangles. Let $j\;$ be a suitable rotation through $120°.\;$ Then the fact that triangles $ABC_{1},\;$ $CAB_{1},\;$ and $BCA_{1}\;$ are equilateral may be expressed as

$\begin{align} (1) & A + jB + j^{2}C_{1} = 0\\ (2) & C + jA + j^{2}B_{1} = 0\\ (3) & B + jC + j^{2}A_{1} = 0 \end{align}$

The center of $\Delta ABC_{1}\;$ is given by $P = (A + B + C_{1})/3,\;$ and similarly for centers $Q\;$ and $R\;$ of triangles $CAB_{1}\;$ and $BCA_{1}:\;$ $Q = (C + A + B_{1})/3\;$ and $R = (B + C + A_{1})/3.\;$ We want to show that $P + jQ + j^{2}R = 0.\;$ Indeed,

$\begin{align} 3(P + jQ + j^{2}R) &= A + B + C_{1} + j(C + A + B_{1}) + j^{2}(B + C + A_{1})\\ &= (B + jC + j^{2}A_{1}) + j(A + jB + j^{2}C_{1}) + j^{2}(C + jA + j^{2}B_{1})\\ &= 0. \end{align}$

### 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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