Napoleon's Theorem: Intermediate Challenge

Background

Napoleon's Theorem reveals a fascinating property involving equilateral triangles constructed on the sides of any triangle.

Part 1: Construction and Proof

Given triangle ABC with vertices:

A(0, 0), B(6, 0), C(2, 4)
  1. Construct and calculate the vertices (D, E, F) of the outward equilateral triangles on each side.
  2. Find the centers (P, Q, R) of these equilateral triangles.
  3. Prove that PQR is equilateral using:
    • Complex numbers method
    • Vector algebra
    • Classical geometric proof

Part 2: Analysis

  1. Express the area of PQR in terms of:
    • The side lengths of ABC
    • The area of ABC
  2. Prove that the ratio Area(PQR)/Area(ABC) is constant for all triangles ABC.
  3. Find the exact value of this ratio.

Part 3: Extension

Consider the inner Napoleon triangles (constructed inward) and prove that:

Area(outer Napoleon triangle) - Area(inner Napoleon triangle) = Area(ABC)
To find vertices D, E, F, use rotation matrices with angle 60°.
The centroid of an equilateral triangle is located at (x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3.
Consider using complex numbers: multiply vectors by e^(iπ/3) for 60° rotations.