# Triangles, Ellipses, and Cubic Polynomials

@article{Minda2008TrianglesEA, title={Triangles, Ellipses, and Cubic Polynomials}, author={David Minda and Steve Phelps}, journal={The American Mathematical Monthly}, year={2008}, volume={115}, pages={679 - 689} }

(2008). Triangles, Ellipses, and Cubic Polynomials. The American Mathematical Monthly: Vol. 115, No. 8, pp. 679-689.

## 22 Citations

Duality and Inscribed Ellipses

- Mathematics
- 2015

We give a constructive proof for the existence of inscribed families of ellipses in triangles and convex quadrilaterals; a unique ellipse exists in a convex pentagon. The techniques employed are…

A Generalization of the Steiner Inellipse

- MathematicsAm. Math. Mon.
- 2020

The conics passing through six points on the sides of a triangle equidistant from the corresponding midpoints are classified using tools from synthetic and real algebraic geometry.

Polynomials, Ellipses, and Matrices: Two Questions, One Answer

- MathematicsAm. Math. Mon.
- 2011

It is shown that the answer to both questions for points a1, a2 in the unit disc is a very simple: if and only if 2❘a1a2❘ = ❘ a1 + a2 ❘.

Lines of Best Fit for the Zeros and for the Critical Points of a Polynomial

- MathematicsAm. Math. Mon.
- 2011

Abstract Combining results presented in two papers in this Monthly yields the following elementary result. Any line of best fit for the zeros of a polynomial is a line of best fit for its critical…

Four Theorems with Their Foci on Ellipses

- MathematicsAm. Math. Mon.
- 2019

Though Siebeck’s theorem is a geometric statement about complex functions, it is used linear algebra and the numerical range of a matrix to provide a proof of the theorem.

Quartic Coincidences and the Singular Value Decomposition

- Mathematics
- 2013

Summary The singular value decomposition is a workhorse in many areas of applied mathematics and the insights it gives to linear transformations is beautiful. Using the geometry given by the SVD, we…

An Area Inequality for Ellipses Inscribed in Quadrilaterals

- Mathematics
- 2009

If E is any ellipse inscribed in a convex quadrilateral, D, then we prove that Area(E)/Area(D) is less than or equal to pi/4, and equality holds if and only if D is a parallelogram and E is tangent…

The Geometry of Cubic Polynomials

- Philosophy
- 2014

Summary We study the critical points of a complex cubic polynomial, normalized to have the form p(z) = (z - 1)(z - r1)(z - r2) with |r1| = 1 |r2|. If Tγ denotes the circle of diameter passing through…

A Geometric Proof of the Siebeck–Marden Theorem

- MathematicsAm. Math. Mon.
- 2017

This work provides a new direct proof of a general form of the result of Siebeck and Marden that every inellipse for a triangle is uniquely related to a certain logarithmic potential via its focal points.

Visualizing Marden's theorem with Scilab

- MathematicsArXiv
- 2015

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.

## References

SHOWING 1-10 OF 13 REFERENCES

Geometry of Polynomials

- Mathematics
- 1970

During the years since the first edition of this well-known monograph appeared, the subject (the geometry of the zeros of a complex polynomial) has continued to display the same outstanding vitality…

An Introduction to Complex Function Theory

- Mathematics
- 1995

This book provides an introduction to the theory of analytic functions of a single complex variable. While presupposing in its readership a degree of mathematical maturity, it insists on no formal…

An Elementary Proof of Marden's Theorem

- MathematicsAm. Math. Mon.
- 2008

A complete proof of Marden’s Theorem is obtained, requiring very little beyond standard topics from undergraduate mathematics, and spanning analytic geometry, linear algebra, complex analysis, calculus, and properties of polynomials.

MATH

- Biology
- 1992

It is still unknown whether there are families of tight knots whose lengths grow faster than linearly with crossing numbers, but the largest power has been reduced to 3/z, and some other physical models of knots as flattened ropes or strips which exhibit similar length versus complexity power laws are surveyed.

edu REFERENCES 1. I. Grattan-Guiness, A limerick retort

- this MONTHLY
- 2005

Prussian Academy of Sciences

- Prussian Academy of Sciences

A limerick retort

- MONTHLY
- 2005