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 March 21st, 2012, 01:56 AM #1 Senior Member   Joined: Jan 2007 From: India Posts: 161 Thanks: 0 Are 3 points required to define a circle!? Hi All, I read somewhere a while ago that we need 3 points to define a circle. I understand 2 points define a line. But cant a circle also be defined with 2 points, say the center point and one more point on the circumference. Please clarify if I am missing something. Cheers , Arun
 March 21st, 2012, 02:43 AM #2 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: Are 3 points required to define a circle!? The way i understand it, the center is not on the circle, when they talk about points defining a unique circle, they mean points on the circle. In this sense, you need 3 NON-COLINEAR points (all 3 not on the same line) to define a circle because if the points are on a straight line then at least 1 point will not be on the circle if you try to draw a circle through 3 colinear points. also, extending your analogy backwards, you can define a unique line using only 1 point and the slope but if you want to get a unique line using points on the line you need 2. Similarly, if you want to get a unique circle using points on the circle you need 3 NON-COLINEAR points.
 March 21st, 2012, 10:21 AM #3 Senior Member   Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11 Re: Are 3 points required to define a circle!? so to make his answer easier. if you have 2 points you can take them both as the center. so thats why you need 3 then you can't chose which of them is the center
March 21st, 2012, 10:58 AM   #4
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Re: Are 3 points required to define a circle!?

Quote:
 Originally Posted by gelatine1 so to make his answer easier. if you have 2 points you can take them both as the center. so thats why you need 3 then you can't chose which of them is the center
Uh... no.

x^2 + y^2 + ax + by + c = 0 is an equation of a circle.

We can solve the system of equations given by three points, to determine values of a,b, and c.

March 21st, 2012, 04:57 PM   #5
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Re: Are 3 points required to define a circle!?

Quote:
 Originally Posted by arun But cant a circle also be defined with 2 points, say the center point and one more point on the circumference.
I don't fully understand the current comments. I've yet to take Geo, but. . .

It makes perfect sense that a center and a point on your circle can define the circle.

Consider,

Define O, the center of the circle, as (x,y). Consider the point on the circle (n,m). The radius of the circle, r, is given by the distance between the center and the point on the circle. This would be simply sqrt((n-x)^2+(y-m)^2). Given this amount, r, we can determine any other point on the circle simply by choosing a particular angle and calculating what point is the distance of r from the center at that angle. (This last sentence was a slight abuse of language.)

I don't feel like going into the details too much, but Arun makes a perfectly valid point to me. Does anyone disagree? If so, why?

March 21st, 2012, 05:48 PM   #6
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Re: Are 3 points required to define a circle!?

Quote:
 Originally Posted by CherryPi It makes perfect sense that a center and a point on your circle can define the circle.
[color=#000000]I think[/color] [color=#FF0000]TheChaz[/color] [color=#000000]answered it! You have to consider randomness too, what you do is to name a point (not random) "a center" pick another one (not random again) and name it "point of the circumference" .[/color]

Quote:
 Originally Posted by TheChaz Uh... no. x^2 + y^2 + ax + by + c = 0 is an equation of a circle. We can solve the system of equations given by three points, to determine values of a,b, and c.

 March 22nd, 2012, 01:12 AM #7 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: Are 3 points required to define a circle!? A circle is the set of all points equidistant from a fixed point called the center. The center is not part of the circle. Yes, TheChaz is right about his system of equations but the points chosen must be on the circumference of the sought after circle to be determined. Yes, if you give a center and a point on the circumference a unique circle is defined, also you can give a center and radius to define a unique circle, however if you are given only points on the circumference then you need minimum 3 NON-COLINEAR points to define a unique circle.
 March 22nd, 2012, 03:19 PM #8 Senior Member   Joined: Jul 2011 Posts: 245 Thanks: 0 Re: Are 3 points required to define a circle!? And the goal of this website was just accomplished. Good job, everyone. Our unique perspectives are invaluable to us.
March 22nd, 2012, 03:30 PM   #9
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Re: Are 3 points required to define a circle!?

Quote:
 Originally Posted by agentredlum A circle is the set of all points equidistant from a fixed point called the center. The center is not part of the circle. Yes, TheChaz is right about his system of equations but the points chosen must be on the circumference of the sought after circle to be determined. Yes, if you give a center and a point on the circumference a unique circle is defined, also you can give a center and radius to define a unique circle, however if you are given only points on the circumference then you need minimum 3 NON-COLINEAR points to define a unique circle.
I should have emphasized that the "center" is not part of the circle (except in the degenerate case where r = 0).

Maybe someone with a bunch of time on their hands would like to show that the system

$x_1^2 + y_1^2 + ax_1 + by_1 + c= 0$

$x_2^2 + y_2^2 + ax_2 + by_2 + c= 0$

$x_3^2 + y_3^2 + ax_3 + by_3 + c= 0$

Has solutions only when $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ are non-colinear...

Not me!

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# what is the minimum number of points required to determine a unique circle?

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