Why do we need perspective division?

Question

I know perspective division is done by dividing x,y, and z by w, to get normalized device coordinates. But I am not able to understand the purpose of doing that. Also, does it h

Answer 1

To really distill it to the basic concept, and why the op is division (instead of e.g. square root or some such), consider that an object twice as far should appear with dimensions exactly one half as large. Obtain 1/2 from 2 by... division.

There are many geometric ways to arrive at the same conclusion. A diagram serves as visual proof for this, really.