An Elementary Proof of Marden's Theorem

  title={An Elementary Proof of Marden's Theorem},
  author={Dan Kalman},
  journal={The American Mathematical Monthly},
  pages={330 - 338}
  • D. Kalman
  • Published 1 April 2008
  • Mathematics
  • The American Mathematical Monthly
I call this Marden’s Theorem because I first read it in M. Marden’s wonderful book [6]. But this material appeared previously in Marden’s earlier paper [5]. In both sources Marden attributes the theorem to Siebeck, citing a paper from 1864 [8]. Indeed, Marden reports appearances of various versions of the theorem in nine papers spanning the period from 1864 to 1928. Of particular interest in what follows below is an 1892 paper by Maxime Bocher [1]. In his presentation Marden states the theorem… 

Visualizing Marden's theorem with Scilab

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.

Another Application of Siebeck's Theorem

If z1, z2, and z3 are three points in C and p(z) = (z − z1), then the zeros of p′ are the foci of the Steiner ellipse for z 1,Z2, z3, as follows.

Pitot's theorem, dynamic geometry and conics

It is well known that a convex quadrilateral is a cyclic quadrilateral if, and only if, the sum of each pair of opposite angles is π. This result (which gives a necessary and sufficient condition for

The Geometry of Cubic Polynomials

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

Four Theorems with Their Foci on Ellipses

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.

A Geometric Proof of the Siebeck–Marden Theorem

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.

A Final Report on the SAIF Grant , 2011-2012

My research plan was to study alpha-regular stick unknots. After an exhaustive study of the literature, I revisited the case of an even number of sticks and came up with a new approach using points

Geometry of a Class of Generalized Cubic Polynomials

This paper studies a class of generalized complex cubic polynomials of the form p(z)=(z-1)(z-r_1)^k(z-r_2)^k where r_1 and r_2 lie on the unit circle and k is a natural number.  We completely

Leaky roots and stable Gauss-Lucas theorems

ABSTRACT Let be a polynomial. The Gauss-Lucas theorem states that its critical points, , are contained in the convex hull of its roots. In a recent quantitative version, Totik shows that if almost

Položaj nultočaka polinoma

By Rolle’s Theorem, any segment whose endpoints are mutually distinct real roots of a polynomial p : R→ R contains at least one stationary point of the polynomial p. Complex analogues of this theorem



The Most Marvelous Theorem in Mathematics

Picture the graph of a cubic, with three x intercepts (also known as the roots). We know from calculus that the derivative of the cubic will have two roots, one between each pair of consecutive

A note on the zeros of the sections of a partial fraction

where z\, z2 and Zz are three distinct, noncollinear points, lie at the foci of the conic which touches the line segments (z2, Zz), ( 3, Zi) and (01, 02) in the points fi, f2 and f3 that divide these

On certain polar curves with their application to the location of the roots of the derivatives of a rational function

The question asked by Professor Maxime Bochert in a discussion concerninig certain theorems on the roots of the derivative of a polynomial, "Could not the first of these propositions be brought into

Ueber eine neue analytische Behandlungweise der Brennpunkte

  • J. Reine Angew. Math
  • 1864

Geometry of Polynomials, Math. Surveys no

  • Geometry of Polynomials, Math. Surveys no
  • 1966

and R

  • F. Davis, Geometrical Conics, MacMillan, New York,
  • 1894