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A {\\em $k$-collinearity} is a pair $(L,t)$ of a line $L$ and a time $t$ such that $L$ contains at least $k$ points at time $t$, the points along $L$ do not all coincide, and not all of them are collinear at all times. We show that, if the points move with constant velocity, then the number of 3-collinearities is at most $2\\binom{n}{3}$, and this bound is tight. There are $n$ points having $\\Omega(n^3/k^4 + n^2/k^2)$ distinct $k$-collinearities. 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