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Finding the Center of a Circle from Three Points


Date: 05/22/2000 at 10:18:50
From: Christian Furst
Subject: Center point of circle

I have the coordinates of three ordered points on a circle. I want to 
find a way to define the circle's center. The purpose is to find a way 
to draw the part of the circle that is connecting the three points (by 
knowing the center and radius).


Date: 05/22/2000 at 15:34:50
From: Doctor Rob
Subject: Re: Center point of circle

Thanks for writing to Ask Dr. Math, Christian.

Let the equation of the circle be

     (x-h)^2 + (y-k)^2 = r^2,

and substitute the three known points, getting 3 equations in 3 
unknowns h, k, and r:

     (x1-h)^2 + (y1-k)^2 = r^2
     (x2-h)^2 + (y2-k)^2 = r^2
     (x3-h)^2 + (y3-k)^2 = r^2

which you can solve simultaneously. First subtract the third equation 
from the other two, thus eliminating r^2, h^2, and k^2. That will 
leave you with 2 simultaneous linear equations in h and k to solve. 
This you can do as long as the 3 points are not collinear. Then those 
values of h and k can be used in the first equation to find the 
radius:

     r = sqrt[(x1-h)^2 + (y1-k)^2].

Example: Suppose a circle passes through the points (4,1), (-3,7), and 
(5,-2). Then we know that:

     (h-4)^2 + (k-1)^2 = r^2
     (h+3)^2 + (k-7)^2 = r^2
     (h-5)^2 + (k+2)^2 = r^2

Subtracting the first from the other two, you get:

     (h+3)^2 - (h-4)^2 + (k-7)^2 - (k-1)^2 = 0,
     (h-5)^2 - (h-4)^2 + (k+2)^2 - (k-1)^2 = 0,

     h^2 + 6h + 9 - h^2 + 8h - 16 + k^2 - 14k + 49 - k^2 + 2k - 1 = 0
     h^2 - 10h + 25 - h^2 + 8h - 16 + k^2 + 4k + 4 - k^2 + 2k - 1 = 0

     14h - 12k + 41 = 0
     -2h +  6k + 12 = 0

     10h +  65 = 0
     30h + 125 = 0

     h = -13/2
     k = -25/6

Then

     r = sqrt[(4+13/2)^2 + (1+25/6)^2]
       = sqrt[4930]/6

Thus the equation of the circle is:

     (x+13/2)^2 + (y+25/6)^2 = 4930/36

Understood?

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Basic Algebra
High School Conic Sections/Circles
High School Coordinate Plane Geometry
High School Geometry

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