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Math Help - Simplest Formula to Find Equation of A Circle from 3 Points Given

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    Simplest Formula to Find Equation of A Circle from 3 Points Given

    I need a simplest and 100% correct formula (or method) to find equation of a circle with 3 points given.
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    Quote Originally Posted by samtcs View Post
    I need a simplest and 100% correct formula (or method) to find equation of a circle with 3 points given.
    Given that (x_1, y_1), (x_2,y_2), and (x_2,y_3) are points on the plane, then ((x_1+x_2)/2, (y_1+y_2)/2) and ((x_1+x_3)/2, (y_1+y_3)/2) are midpoints of chords in the circle. The two chords have slopes (y_1- y_2)/(x_1- x_2) and (y_1- y_3)/(x_1-x_3) respectively so that -(x_1-x_2)/(y-1-y_2) and -(x_1-x_3)/(y_1-y_3) are the slopes of the perpendicular lines.

    That means that y= -(x_1-x_2)/(y_1-y_2)(x- x_1)+ y_1 and y= -(x_1- x_3)/(y_1-y_2)(x-x_1)+ y_1 are equations of the radii through the midpoints and so intersect the center, (x_0,y_0). Solve those two equations simultaneously to find x= x_0 and y= y_0. Of course, once you have that, the radius of the circle is given by \sqrt{(x_1-x_0)^2+ (y_1- y_0)^2} and then the equation of the circle is (x- x_0)^2+ (y_1- y_0)^2= r^2.
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