Visualizing Marden's theorem with Scilab

TL;DR

This paper demonstrates how to visualize Marden's theorem using Scilab, illustrating the relationship between triangle vertices in the complex plane and the zeros of related polynomials.

Contribution

It provides a method to visualize Marden's theorem and compute Steiner ellipses in the complex plane using Scilab, an open-source software.

Findings

Successful visualization of Marden's theorem with arbitrary complex points

Calculation and plotting of Steiner ellipses for given triangles

Open source implementation in Scilab facilitates educational and research use

Abstract

A theorem which is named after the American Mathematician Moris Marden states a very surprising and interesting fact concerning the relationship between the points of a triangle in the complex plane and the zeros of two complex polynomials related to this triangle: "Suppose the zeroes z1, z2, and z3 of a third-degree polynomial p(z) are non-collinear. There is a unique ellipse inscribed in the triangle with vertices z1, z2, z3 and tangent to the sides at their midpoints: the Steiner in-ellipse. The foci of that ellipse are the zeroes of the derivative p'(z)." (Wikipedia contributors, "Marden's theorem", http://en.wikipedia.org/wiki/Marden%27s_theorem). This document describes how Scilab, a popular and powerful open source alternative to MATLAB, can be used to visualize the above stated theorem for arbitrary complex numbers z1, z2, and z3 which are not collinear. It is further demonstrated how the equations of the Steiner ellipses of a triangle in the complex plane can be calculated and plotted by applying this theorem.