How can I determine the center of a village?

Since an Iron Golem’s spawning zone is a 16×16×6 area at the center point of a village, I want to build a structure around this area. Sometimes I want to build a grinder there, other times I want to build something that looks like a golem forge to make it look realistic instead of them just randomly appearing out of thin air. To do this, I need to determine the center of the village.

How can I determine the center of a village regardless of whether it is one built from scratch or a generated one that is naturally spawned? Also, for generated villages, is the center determined before or after the village is built? That is, is the center point chosen then structures built around it, or are the structures generated first then the center calculated from them afterwards?

(I don’t think it matters, but I’m using version 1.7.2.)

There are two village centers to consider here; one on creation, and the one you actually care about.

On world generation, the village's well is considered the center (or starting point) of the village. It is the first structure placed, and the entire village construction is based upon the well's location.

The center of the village you actually care about is the centroid of all the doors in the immediate vicinity. If you took all the coordinates of all the doors in your village and averaged them together, this would be the area where golems would spawn.

For additional details on the logic of the center of the village, read this section in the Minecraft Wiki.