There is a set of 39 distinct points on a plane with the following…

To count triangles with positive area from a set of points, choose any 3 points that are not all collinear, since three collinear points give zero area and cannot form a triangle. When one collinear subset is known, split the count of valid triangles by how many of the 3 chosen vertices come from that subset, excluding the case where all 3 come from it.

Here, of the 39 points, 8 lie in the collinear subset A and the remaining 31 points are non-collinear (no 3 of them, and no other combination outside A, are collinear).

• 1 vertex from the 8 collinear points and 2 vertices from the 31 non-collinear points

• 2 vertices from the 8 collinear points and 1 vertex from the 31 non-collinear points

• 3 vertices from the 31 non-collinear points

Adding the combinations for these three valid cases gives the total number of triangles with positive area:

1. 8C1 × 31C2 = 8 × 465 = 3720

2. 8C2 × 31C1 = 28 × 31 = 868

3. 31C3 = 4495

4. Total = 3720 + 868 + 4495 = 9083

As an independent check, count all triples from the full 39 points and remove only the ones fully inside the 8-point collinear group (which cannot form a triangle): 39C3 − 8C3 = 9139 − 56 = 9083, the same result.

So, 9083 distinct triangles with positive area can be formed.