Let $m,n\geq 2$ be positive integers and $S\subseteq [1,m]\times [1,n]$ be a set of lattice points. Prove that if \[|S|\geq m+n+\bigg\lfloor\frac{m+n}{4}-\frac{1}{2}\bigg\rfloor\]then there exists a circle which passes through at least four distinct points of $S.$