3-ANGLES

I counted all the triangles and solved the question.
There are 9 cases according to me :
$Case 1$ : All the small fragments : $\boxed{12}$

$Case 2$ : Two at a time : $\boxed{6}$

$Case 3$ : Half at a time = $2 \times 3$ : $\boxed{6}$

$Case 4$ :Half at a time = $2 \times 3$ : $\boxed{6}$

$Case 5$ : One-third a time : $\boxed{3}$

$Case 6$ :Half at a time from each vertex = $2 \times 3$ : $\boxed{6}$

$Case 7$ : Just parts : $\boxed{6}$

$Case 8$ : Big triangle as a whole : $\boxed{1}$

$Case 9$ : Small triangle as a whole : $\boxed{1}$

Thus total number of triangles : $\boxed{47}$