## Tuesday, January 08, 2008

### Equation for a circle

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

Proof:

(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

QED

Anonymous said...

The equation in the postulate contradicts the proof. In the postulate you say d =, In the proof you say r =.

Larry Freeman said...

Hi Anonymous,

I'm sorry if this point is not clear.

d is a function that can be used between any two points.

r refers to two points that describe a circle.

For the proof, I am applying the formula for d to the two points that make up r.