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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 NONCOLINEAR 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 NONCOLINEAR 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  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Are 3 points required to define a circle!? Quote:
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  
Senior Member Joined: Jul 2011 Posts: 245 Thanks: 0  Re: Are 3 points required to define a circle!? Quote:
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((nx)^2+(ym)^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  
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Are 3 points required to define a circle!? Quote:
Quote:
 
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 NONCOLINEAR 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  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Are 3 points required to define a circle!? Quote:
Maybe someone with a bunch of time on their hands would like to show that the system Has solutions only when are noncolinear... Not me!  

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