64 triangles
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The answer depends on how many points there are and how many triplets are non-collinear.
Infinitely many planes may contain the same three collinear points if the planes all intersect at the same line.
There are an infinite number of any kind of points in any plane. But once you have three ( 3 ) non-collinear points, you know exactly which plane they're in, because there's no other plane that contains the same three non-collinear points.
5 triangles are formed.
5 equilateral triangles.