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My solution. Took much longer than usual to solve: I had the right picture but with 540 triangles. Had to read others' comments to come up with the right final filer.
Here's my solution:
Fun challenge. I got 540 triangles, probably because some very thin triangles should be counted as collinear.
My solution is oh so very long and clunky, nowhere near as elegant as @TerryT 's solution...
I also went with looking at direction between the points and thus ended up with an additional 3 tiny triangles.
I took the non-colinear points to mean no points of the triangle share a lat/long - so I ended up with 184 unique triangles, kind of mesmerizing.
I used "No equal degree of line" rule in ruling out col-linear dots and ended up with 535 triangles.
I suppose it's the spatial projection issue and this is within acceptance range?
Here's my solution