How to prove 1/f= 1/u +1/v?

I think the derivation is geometrical - you know the ray going through a focus is parallel on the other side, and the ray going through the center doesn't bend. You end up with a whole bunch of triangles you can compare. Properly applied geometry gives you the formula.

This link might be helpful:

 

http://www.saburchill.com/physics/chapters3/0008.html

 

As you said, it's a geometrical proof.

There is another form of this equation. u * v = f^2

 

However both u and v are measured from the focal point.