# Isosceles trapezoids, perpendicular bisectors, and few distinct distances

## Ben Lund, Rutgers University

## March 10, 2015

Erdős famously conjectured that every set of $n$ points in the real plane
determines $\Omega(n/\sqrt{\log n})$ distinct distances. He further asked
if a set of points that determines $O(n/\sqrt{\log n})$ distances must
have some special structure; in particular, is there a line that
contains contains $n^\varepsilon$ points, for some $\varepsilon>0$?

I will discuss a new upper bound on the number of isosceles trapezoids
determined by a set of $n$ points, no more than $k$ of which lie on any
circle or line, and the application of this bound to the
aforementioned question of Erdős. I will also mention a result and an
open problem on the number of distinct perpendicular bisectors
determined by a set of points.

Based on joint work with Adam Sheffer and Frank de Zeeuw.