Constructing borders between points

I found if you have a collection of randomly distributed points a way to logically construct borders between them. Say these are the points:

First you construct lines halfway between the points like so, these four lines are halfway between the point in the middle and each of the surrounding points. Basically you draw a circle with the center at one point, and extending to another point and do the same starting at the second point, then connect a line between where those circles intersect. They form four new points at intersections of lines I'll call a,b,c,d that form a four sided figure:

When you continue on this process you will notice that the line halfway between two of the other points crosses through the four corners of the polygon you had already made. 

So then cleaning up you end up with: