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Find Circle Center and Radius

Date: 09/21/2001 at 08:51:17
From: Alex
Subject: Geometry

If you are given three sets of (x,y) coordinates that lie on the
circumference of a circle, how do you figure out the center and radius
of the circle?

Date: 09/21/2001 at 13:43:40
From: Doctor Rob
Subject: Re: Geometry

Thanks for writing to Ask Dr. Math, Alex.

You can start with the three-point form of the equation of the circle,
which can be found on the following page from our Frequently Asked
Questions (FAQ):   

   | x^2+y^2  x  y  1|
   |x1^2+y1^2 x1 y1 1|
   |x2^2+y2^2 x2 y2 1| = 0.
   |x3^2+y3^2 x3 y3 1|

Here (x1,y1), (x2,y2), and (x3,y3) are the three given points. Expand
this 4-by-4 determinant, complete the squares on x and y, and 
transform it into the center-radius 1form

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

Then the center is (h,k) and the radius is r.

Explicit expressions for h, k, and r^2 are as follows:

       |x1^2+y1^2 y1 1|
       |x2^2+y2^2 y2 1|
       |x3^2+y3^2 y3 1|
   h = ----------------,
           |x1 y1 1|
         2*|x2 y2 1|
           |x3 y3 1|

       |x1 x1^2+y1^2 1|
       |x2 x2^2+y2^2 1|
       |x3 x3^2+y3^2 1|
   k = ----------------.
           |x1 y1 1|
         2*|x2 y2 1|
           |x3 y3 1|

                     |x1 y1 x1^2+y1^2|
                     |x2 y2 x2^2+y2^2|
                     |x3 y3 x3^2+y3^2|
   r^2 = h^2 + k^2 + -----------------.
                         |x1 y1 1|
                         |x2 y2 1|
                         |x3 y3 1|

In expanded form, these are:

        (x1^2+y1^2)(y2-y3) + (x2^2+y2^2)(y3-y1) + (x3^2+y3^2)(y1-y2)
    h = ------------------------------------------------------------
              2(x1y2 - x2y1 - x1y3 + x3y1 + x2y3 - x3y2)

        (x1^2+y1^2)(x3-x2) + (x2^2+y2^2)(x1-x3) + (x3^2+y3^2)(x2-x1)
    k = ------------------------------------------------------------
              2(x1y2 - x2y1 - x1y3 + x3y1 + x2y3 - x3y2)

                            (x1^2+y1^2)(x2y3-x3y2) + 
                            (x2^2+y2^2)(x3y1-x1y3) + 
    r^2 = h^2 + k^2 + ---------------------------------------
                      x1y2 - x2y1 - x1y3 + x3y1 + x2y3 - x3y2

- Doctor Rob, The Math Forum   

Date: 05/17/2002 at 10:31:05
From: Michael Greene
Subject: Equation for a circle given (x,y) of 3 points

I have posed this question to geometry and Algebra 2 classes for the 
past 3-4 years, and the geometric solution may be simpler for many 
students who recall the geometry and may not be as comfortable with 
systems of equations:

Consider A(-2,1) B(2,3) C(0,-5)

a) The center of a circle may be found by the intersection of the 
   perpendicular bisectors of 2 chords.

b) Find the slopes and midpoints of 2 chords:
     AB: (0,2) m = 1/2
     BC: (1,-1) m = 4

c) Find the slopes of the perpendicular lines.
d) Find the equations of the perpendicular bisectors:
    Through AB, y = -2x+2
    Through BC, y = (-1/4)x  -3/4

e) Find the intersection of those lines
    -2x +2 = (-1/4)x -3/4
    -8x +8 = -x -3
    -7x = -11    
     x = 11/7 and y = -8/7

This isn't so much a question as an alternate solution, but I wanted 
to give something back to the Math Forum after referring some of my 
students to the archives and finding new ideas for our classes.

Mike Greene
Associated Topics:
High School Conic Sections/Circles
High School Coordinate Plane Geometry
High School Geometry

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