Inspired by chess board

No. Of rectangles is ((n+1)C2)^2....hence no . Of right angled triangle is 4× no. Of rectangles in the n×n chessboard.

There are 9 horizontal & vertical lines in the chess board consisting of 9 points.

For the horizontal line 'a' we have ,

We can select any two points in $\binom{9}{2}$ ways. Observe that for a pair of points , there are two reflecting points in it's adjacent horizontal line.

Suppose if we choose the lines 'a' & 'b' , we can have a reflecting point on 'b' for each of the points on 'a'.

We make 2 right angled triangles for any pair of points in a line with respect to another line. So number of right angled triangles for a line with respect to another line is $\binom{9}{2}\times 2=72$ .

There are 9 lines , so number of right triangles for a line with respect to all other 8 lines = $72\times8=576$

With the help of 1 line we can build up $\text{576}$ right triangles. For all 9 lines we can build $576\times9=\boxed{5184}$

$\text{Bonus : Generalize for a m }\times\text{ n board}$