Gridlines

How many triples of points are there in this 4x4 array that lie on a straight line?
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Problem

 
There are lots of ways of choosing three dots from this $4$ by $4$ array.
 
Image
Gridlines


How many triples of points are there where all three lie on a straight line (not necessarily equally spaced)?

If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.