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The One-Seventh Area Triangle in the Complex Plane: Proving the Proportionality Using Features of the Complex Number System

Mathew Miltonhardy

TL;DR

The paper proves that the inner triangle formed by cevians to the points one-third along the sides of an outer triangle in the complex plane has area equal to one-seventh of the outer triangle, using a purely algebraic approach. It fixes a canonical configuration with $p=0$, $q$ real, and $r=x+iy$, computes the outer area $A_{pqr}$, derives edge points $P,Q,R$, and solves for the inner vertices $\alpha,\beta,\gamma$ via a conjugate-trick, obtaining $A_{\alpha\beta\gamma}=\frac{1}{7}A_{pqr}$. The explicit coordinates yield $\alpha=\frac{3}{7}P$, $\beta=q+\frac{3}{7}(Q-q)$, and $\gamma=\frac{6}{7}P$, confirming the proportionality. By area invariance under translations and rotations, the result extends to all triangles in the complex plane, providing a robust algebraic proof of Feynman’s triangle.

Abstract

This paper explores and proves the one-seventh area triangle using a purely algebraic approach as opposed to a geometric one. A triangle set purely in the complex plane is used so that we can utilise features of the complex number system to determine areas. The area of a triangle in determinant form is derived using these features, and a rigorous proof of the one-seventh area triangle follows.

The One-Seventh Area Triangle in the Complex Plane: Proving the Proportionality Using Features of the Complex Number System

TL;DR

The paper proves that the inner triangle formed by cevians to the points one-third along the sides of an outer triangle in the complex plane has area equal to one-seventh of the outer triangle, using a purely algebraic approach. It fixes a canonical configuration with , real, and , computes the outer area , derives edge points , and solves for the inner vertices via a conjugate-trick, obtaining . The explicit coordinates yield , , and , confirming the proportionality. By area invariance under translations and rotations, the result extends to all triangles in the complex plane, providing a robust algebraic proof of Feynman’s triangle.

Abstract

This paper explores and proves the one-seventh area triangle using a purely algebraic approach as opposed to a geometric one. A triangle set purely in the complex plane is used so that we can utilise features of the complex number system to determine areas. The area of a triangle in determinant form is derived using these features, and a rigorous proof of the one-seventh area triangle follows.
Paper Structure (12 sections, 26 equations, 3 figures)

This paper contains 12 sections, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Area of the Outer Triangle
  3. Determining Coordinates
  4. The Edges
  5. The Inner Vertices
  6. Determining $\alpha$
  7. Determining $\beta$
  8. Determining $\gamma$
  9. Area of the Inner Triangle
  10. Conclusion
  11. Acknowledgements
  12. Further Research

Figures (3)

  • Figure 1: Triangle formed by vertices in the complex plane where $\boldsymbol{p}$ is at the origin, $\boldsymbol{q}$ is purely real, and $\boldsymbol{r}$ is any point in the complex plane. Cevians are drawn from each vertex to the point one-third along the opposite side in a clockwise direction.
  • Figure 2: Triangle formed by vertices in the complex plane where $\boldsymbol{p}$ is at the origin, $\boldsymbol{q}$ is purely real, and $\boldsymbol{r}$ is any point in the complex plane. The height of the triangle is labelled $\boldsymbol{h}$.
  • Figure 3: Triangle formed by vertices in the complex plane where $\boldsymbol{p}$, $\boldsymbol{q}$, and $\boldsymbol{r}$ are any points in the complex plane. The height of the triangle is labelled $\boldsymbol{h}$, and the angle $\angle \boldsymbol{p}\boldsymbol{q}\boldsymbol{r}$ is labelled $\theta$.