Napoleon's Theorem

Let the original triangle be ABC with equilateral triangle ABC₁, CAB₁, and BCA₁ 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^o. Then the fact that triangles ABC₁, CAB₁, and BCA₁ are equilateral may be expressed as

The center of ABC₁ is given by P = (A + B + C₁)/3, and similarly for centers Q and R of triangles CAB₁ and BCA₁: Q = (C + A + B₁)/3 and R = (B + C + A₁)/3. We want to show that P + jQ + j²R = 0. Indeed,

Napoleon's Theorem

• A proof by tessellation
• A proof with complex numbers
• A second proof with complex numbers
• Two proofs
• Inscribed angles
• Douglas' Generalization
• Napoleon's Theorem by Transformation
• A Generalization
• Napoleon's Propeller
• Fermat point
• Kiepert's theorem
• Lean Napoleon's Triangles

Copyright © 1996-2008 Alexander Bogomolny