USAMO 2014 — Days 1 and 2 Question 3

Prove that there is an infinite set of points $\{P_n:n\in\mathbb Z\}$ in the plane such that, for any three distinct integers $a,b,c$, the points $P_a,P_b,P_c$ are collinear if and only if $a+b+c=2014$.

Proposed by Sam Vandervelde (as credited in the supplied compilation).