On the maximum number of isosceles right triangles in a finite point set

Ábrego, Bernardo; Fernández, Silvia; Roberts, David

doi:10.2140/involve.2011.4.27

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On the maximum number of isosceles right triangles in a finite point set

Bernardo M. Ábrego, Silvia Fernández-Merchant and David B. Roberts

Vol. 4 (2011), No. 1, 27–42

DOI: 10.2140/involve.2011.4.27

Abstract

Let $Q$ be a finite set of points in the plane. For any set $P$ of points in the plane, $S_{Q} (P)$ denotes the number of similar copies of $Q$ contained in $P$ . For a fixed $n$ , Erdős and Purdy asked for the maximum possible value of $S_{Q} (P)$ , denoted by $S_{Q} (n)$ , over all sets $P$ of $n$ points in the plane. We consider this problem when $Q = △$ is the set of vertices of an isosceles right triangle. We give exact solutions when $n \leq 9$ , and provide new upper and lower bounds for $S_{△} (n)$ .

Keywords

Erdős problems, similar triangles, isosceles right triangles

Mathematical Subject Classification 2000

Primary: 52C10

Secondary: 05C35

Milestones

Received: 14 January 2010

Revised: 27 February 2011

Accepted: 27 February 2011

Published: 22 September 2011

Communicated by Kenneth S. Berenhaut

Authors

Bernardo M. Ábrego
Department of Mathematics California State University 18111 Nordhoff Street Northridge, CA 91330-8313 United States
http://www.csun.edu/~ba70714
Silvia Fernández-Merchant
Department of Mathematics California State University 18111 Nordhoff Street Northridge, CA 91330-8313 United States
http://www.csun.edu/~sf70713
David B. Roberts
Department of Mathematics California State University 18111 Nordhoff Street Northridge, CA 91330-8313 United States

Vol. 4, No. 1, 2011