Perspective projection - how to convert coordinates

Question

I'm studying perspective projections and I stumbled upon this concept:

Basically it says that if I have a point (x,y,z) I can project it into my perspective screen (camera space) by doing

x' = x/z
y' = y/z
z' = f(z-n) / z(f-n)

I can't understand why x' = x/z or y' = y/z

Answer 1

Geometrically, it is a matter of similar triangles.

In your diagram, because (x,y,x) is on the same dotted line as (x',y',z'):

triangle [(0,0,0), (0,0,z), (x,y,z)]
  is similar to
triangle [(0,0,0), (0,0,z'), (x',y',z')]

This means that the corresponding sides have a fixed ratio. And, further, the original vector is proportional to the projected vector. Finally, note that the notional projection plane is at z' = 1:

(x,y,z) / z   =  (x',y',z') / z'

  -> so, since z' = 1:
       x'/z' = x' = x/z
       y'/z' = y' = y/z

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[Warning: note that the z' in my answer is different from its occurrence in the question. The question's z' = f(z-n) / z(f-n) doesn't correspond directly to a physical point: it is a "depth value", which is used to do things like hidden surface removal.]