ZIML  Varsity  March  2019)  In  equilateral  triangle ABC,  divide  each  side  evenly  into  8 segments,  and draw lines parallel to the sides to cut the triangle into small triangles,  as shown in the diagram. How many parallelograms are there in this diagram?

In the camp we did 3* (9 choose 4), but if we just take the case where the parallelogram is parallel to AC and BC it seems we can't count the ones with a point on AB by picking 4 out of the 9 points on AB. If we pick 3 points on AB it seems to generate 1 unique parallelogram for each of the 9 choose 3 cases (the middle point extends in both directions diagonally and the other 2 points on either side extend towards the center). So should the answer be 3* ((9 choose 4 )+ (9 choose 3)) instead?

For example: any parallelogram with only one point on AB and consisting of an upside down equilateral triangle under a normal equilateral triangle does not seem to be countable through the 9 choose 4's on any side.