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
3 comments :
The equation in the postulate contradicts the proof. In the postulate you say d =, In the proof you say r =.
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.
If I am misunderstanding your question, please let me know.
-Larry
In my opinion circle making is a easy task just draw a curve in response to a fix line from a central point and all points are the the same distance from the center.
(x-a)2 + (y-b)2 = r2 is standard equation of a circle. where r is radius.
Cone
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