Proof of the 'Mirror Equation'?

Question

1/v + 1/u = 1/f

Answer 1

Let's mark some points:

O - where the object is,
A - point of a mirror hit by the parallel beam from the object,
F - focal point,
C - center point of the mirror
I - image point
B - point of a mirror where parallel beams reflects to the image.

Then from triangles OBI and OFC you get:

IB/CF = (OF + FB)/OF = 1 + FB/OF.......(1)

On the other hand from triangles OFA and BFI you get:

IB/FB = OA/OF or FB/OF = IB/OA.......(2)

Plugging (2) into (1) you get:

IB/CF = 1 + IB/OA or 1/CF = 1/IB + 1/OA.......(3)

Since CF = f, IB = v, OA = u, (3) is your formula.

n.b.: CF = f = 1/2 of the mirror radius.