How to scale a shape made of points?

Question

Say I have a shape like this:

And I want to scale it down to this:

How do I calculate the location of the points (red circles) to do so?

Answer 1

new.x = scale * (old.x - center.x) + center.x
new.y = scale * (old.y - center.y) + center.y

Repeat for each point.