  Let F_q be a finite field with odd q elements. In this article, we prove that
if E \subseteq \mathbb F_q^d, d\ge 2, and |E|\ge q, then there exists a set Y
\subseteq \mathbb F_q^d with |Y|\sim q^d$ such that for all y\in Y, the number
of distances between the point y and the set E is similar to the size of the
finite field \mathbb F_q. As a corollary, we obtain that for each set
E\subseteq \mathbb F_q^d with |E|\ge q, there exists a set Y\subseteq \mathbb
F_q^d with |Y|\sim q^d so that any set E\cup \{y\} with y\in Y determines a
positive proportion of all possible distances. An averaging argument and the
pigeonhole principle play a crucial role in proving our results.
