what is the derivation of mirror formula of concave and convex mirror?

Derivation of formula for convex mirror:

Let AB be an object placed on the principal axis of a convex mirror of focal length f. u is the distance between the object and the mirror and v is the distance between the image and the mirror.

In ABC and A1B1C

<ABC = <A1B1C (right angles)

<ACB = <A1CB1

<CAB = <CA1B (common angle)

ABC is similar to A1B1C

AB/A1B1 = BC/B1C…(1)

similarly DEF issimilar to A1B1F

DE/A1B1 = EF/B1F…(2)

But DE = AB and when the aperture is very small EF = PF

Equation (2) becomes

AB/A1B1 = PF/B1F…(3)

Frm equations (1) and (3) get

PF/B1F = BC/B1C

PF/PF-PB1 = PB + PC/PC - PB1

f/f - v = -u + 2f/2f - v

[PF = f, PB1 = v, PB = u, PC = 2f]

2(2f - v)= (f-v)(2f-u)

i could not write the following 2 steps sorry

-vf + uf + 2 fv -vu=0

fv+uf-vu=0…(4)

Dividing both sides of equation(4) by uvf we get

fv/uvf + uf/uvf - uv/uvf=0

1/u +1/v - 1/f=0

1/u + 1/v = 1/f