Erdős Problem 97: a vertex without four equidistant neighbours

The declaration asserts that for every finite point set $A$ in the Euclidean plane, if $A$ is nonempty and every point of $A$ is absent from the convex hull over $\mathbb{R}$ of the remaining points $A\setminus\{a\}$, then there is at least one point $p\in A$ for which no positive real radius $r$ has at least four points $q\in A$ at distance exactly $r$ from $p$. In expanded form, for that point $p$, every $r>0$ satisfies $\#\{q\in A: \operatorname{dist}(p,q)=r\}<4$. The theorem body is sorry, so this is an asserted target rather than an established result.