How many triagnles are there?

Any three dots may form a triangle except when there are colinear (shown in blue),

No. of non degenerate triangles $= \ ^{36}C_3 - \color{#3D99F6} {14 \times ^6C_3 - 4 \times (^3C_3 + ^4C_3+ ^5C_3)-4 \times 2 \times 4 \times ^3C_3} = \color{#333333} 7140 - 280-60-32= 6768$

Colinear dots (slope =1) $14 \times ^6C_3$ = 6 rows + 6 columns + 2 diagonals each of them has 6 dots.

Colinear dots $4 \times (^3C_3 + ^4C_3+ ^5C_3)$ as shown in figure

Colinear dots (slope =2) $4 \times 2 \times 4 \times ^3C_3$ as shown in figure