Complexity of determining whether three points are collinear from a set of points
There is an $O(n^2)$ algorithm for the more general problem of minimum area triangle, see for example these lecture notes. The problem itself (as well as minimum area triangle) is 3SUM-hard, as shown in a survey of King on 3SUM-hard problems, where the problem is known as 3-points-on-line.

See also this stackoverflow question, where I found the links given above.