Advanced Investigation of Napoleon's Theorem

Extended Napoleon's Theorem Challenge

Consider a complex-analytic approach to Napoleon's Theorem in the context of algebraic geometry and group theory.

Problem Statement:

Let T be any triangle in the complex plane with vertices at z₁, z₂, z₃ ∈ ℂ.

1. Starting with the complex coordinates:

z₁ = -1 + i

z₂ = 2 + 3i

z₃ = 1 - 2i

Tasks:

  1. Prove that if ω = e^(2πi/3) is the cube root of unity, then the centers of the outward equilateral triangles (c₁, c₂, c₃) can be expressed as:
    c_k = z_k + (1-ω)(z_{k+1} - z_k)/3
  2. Show that the difference between the areas of the outer and inner Napoleon triangles is equal to the area of the original triangle multiplied by (2+√3).
  3. Prove that the centers of the Napoleon triangles form a group under complex multiplication when normalized relative to their centroid.
  4. Investigate the relationship between Napoleon's theorem and the Fermat-Weber point by:
    • Expressing the coordinates of the Fermat-Weber point in terms of the centers of the Napoleon triangles
    • Proving that this point minimizes the sum of distances to the vertices
  5. Consider the transformation group G that preserves the Napoleon configuration. Prove it's isomorphic to D₆ (the dihedral group of order 12) and explain its geometric significance.
  6. Bonus Challenge: Extend the theorem to non-Euclidean geometry by:
    • Formulating an analogous statement in hyperbolic geometry
    • Analyzing the limiting behavior as the hyperbolic curvature approaches zero

Additional Considerations: