Postulate 1: The distance d between two points (x1,x2) and (y1,y2)
d = √(x2 - x1)2 + (y2 - y1)2
This postulate assumes all the postulates from Euclid but that works fine for the standard Cartesian coordinates.
Definition 1: A circle
A circle is the set of points equidistant from a given point. This given point is called the center. The distance from the center to any point in the set of points is called the radius.
Theorem 1: Equation of a circle
For any given circle with center at (centerX, centerY) and radius r, the equation is:
(x - centerX)2 + (y - centerY)2 = r2
(1) By Definition 1 above and Postulate 1 above, we have:
r = √(x - centerX)2 + (y - centerY)2
(2) Squaring both sides gives us:
r2 = (x - centerX)2 + (y - centerY)2