connection between an identity and laticial points

Prove that if (p,q)=1 then $\sum_{k=1}^{q-1}\left [ \frac{kp}{q} \right ]=\sum_{k=1}^{p-1}\left [ \frac{kq}{p} \right ]$
Considering a rectangle OABC with A(q,0),B(q,p),C(0,p) let's count the number of points with integer coordinates inside this rectangle:There are $\frac{(p-1)(q-1)}{2}$
On line OB of equation $y=\frac{px}{q}$ we don't have any points of integer coordinates(Prove it)
After that we count the number of points of integer coordinates on the line with x=k under OB,which are $\left [ \frac{kp}{q} \right ]$.If we sum them we have the left side of the identity.
If we replace p with q we have the right side of the identity.

corollary:If (p,q)=d then
Also try to prove that
Now it would it be interesting to find out a generalisation for space and points with integer coordinates in a rectangular cuboid OABCO'A'B'C' where B'(p,q,r),B(p,0,r),...