Define rectangle as two points or origin / size?

This is a question which came up several times in my career, and which splits me and my colleagues into two camps. I thought it might be best answered on this site, once and for all.

Almost all graphics UI libraries implement a Rectangle structure, but the way they do it usually divides along two possible options:

1. The rectangle is defined as two 2D points within the same coordinate system, usually named p0 and p1, or with individual components, as x0, y0, x1, y1.
2. The rectangle is defined as 2D origin and size vectors, usually named position and size, or, individually, x, y, width, height.

Now, say I were to write an UI library, which of the two choices should I take?

Update: The question concerns the implementation, not the interface. Of course the rectangle's interface could well support both approaches, but how would you store the data within the rectangle?