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# Recent Progress in Euclidean Geometry

Workshop on the occasion of the 65th birthday of Professor Paul Yiu

## Titles and Abstracts

>  Robert Bosch: A New Proof of the Erdős-Mordell Inequality

Abstract: In this note we show a new proof of the Erdős-Mordell inequality. The new idea is to consider three interior points to the triangle, then the resulting inequality becomes an Erdős-Mordell inequality when the three before mentioned points coincide.

> Timothy Ford: Sums of Squares in the Field of Rational Functions ℝ(x,y)

Abstract: When Paul Yiu came to our department, his research was on problems related to sums of squares and employed methods that were topological and combinatorial. By limiting ourselves to questions about sums of squares in the field of rational functions ℝ(x,y), this presentation aims to illustrate both the depth and breadth of this interesting subject. We begin by sketching contributions by Hilbert, Landau, Cassels, Ellison and Pfister that prove that the Pythagoras number (the minimum number of squares required to express an arbitrary sum of squares) of the field ℝ(x,y) is four. We end with a theorem due to Colliot-Thélène that shows the existence of many polynomials in ℝ[x,y] that are positive, have even degree at least six, and are not the sum of three squares in ℝ(x,y). The proof is radically different and uses the Noether-Lefschetz Theorem, Galois Cohomology and a Brauer group criterion for sums of three squares.

E-mail: ford@fau.edu

> Lubomir Markov: The Search for Equable and m-Equable Triangles: a Mathematical Delight

Abstract: Triangles with integer sides have fascinated us since ancient times. Let us call an integer-sided triangle equable, if its area is numerically equal to the perimeter, and m-equable, if the area is an integer multiple of the perimeter. It has been known for some time that there are only five equable triangles. In this talk, we will present an algorithm for finding all m-equable triangles for a fixed m.

E-mail: lmarkov@barry.edu

> Petra Surynková: Selected Planar Curves and their Constructions

Abstract: The contribution addresses the geometrical properties of selected planar curves which are determined by moving points or curves in the plane. We will focus on curves such as conic sections, cycloids, epicycloids, conchoids, involutes and evolutes. We will show the examples of the constructions of centers of curvature, osculating circles, cusps, inflection points. Selected planar curves will be also mentioned with respect to geometric problems of antiquity. All constructions will be demonstrated using the dynamic geometry system GeoGebra.

E-mail: Petra.Surynkova@seznam.cz

> Xiao-Dong Zhang: On a Generalization of Bertrand’s Postulate

Abstract: A celebrated theorem in elementary number theory known as Bertrand’s Postulate says that there is at least one prime (number) between n and 2n for every positive integer n ≥ 2. We then wonder if it is also true that there are two primes, or three primes, or more primes between n and 2n. In this talk we outline a history of Bertrand’s Postulate and give the following generalization: Given any positive integer k, there are at least k primes between n and 2n if n ≥ 2k2.

E-mail: xzhang@fau.edu

> Li Zhou: Elementary Extrema in Euclidean Geometry

Abstract: We use examples from problem sections of math journals to illustrate the value of Geometer's Sketchpad in experimenting and discovering solutions.

E-mail: LZhou@polk.edu

> Support: This conference is partially supported by a grant from the Simons Foundation (Award #245848 to the first named organizer)