Putnam 1998 A6

Let $A, B, C$ denote distinct points with integer coordinates in $\mathbb R^2$. Prove that if $$(|AB|+|BC|)^2<8\cdot [ABC]+1$$ then $A, B, C$ are three vertices of a square. Here $|XY|$ is the length of segment $XY$ and $[ABC]$ is the area of triangle $ABC$.