How many triangles are in this situation?

Question

Let's consider a triangle, ABC. Let's fix the points H and K on AB, and the points H' and K' on AC.
Let's link B to H and K, and C to H' and K', by segments: now, several triangles are obtained.
Does anyone know a formula to calculate the exact number of them? Could that formula give the right answer even in case of more than 2 points, fixed on AB and AC?

I apologize. I intended to write "let's link B to H' and K' and C to H and K

Answer 1

Your construction makes triangles and quadrilaterals inside ABC. Here are the numbers of each for one, two, three, and four points, and the general formula for n points.

1 point: 1 quadrilateral, 3 triangles
2 points: 4 quadrilaterals, 5 triangles
3 points: 9 quadrilaterals, 7 triangles
4 points: 16 quadrilaterals, 9 triangles
n points: n^2 quadrilaterals, 2n+1 triangles