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You are given $r$ which is the radius of the two possible circles that can be formed. And you are given two points (coordinates) on their circumferences. How do you compute for the coordinates of the centers of the two circles?

Alex M.
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Mercado
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    draw the line between those points. Then the center is on the line wich is perpendicular to the first line at the mid point of the line. Use this. – L F Sep 06 '15 at 16:00

1 Answers1

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Since the centers are $r$ away from the two given points, $$ (x-x_1)^2+(y-y_1)^2=r^2=(x-x_2)^2+(y-y_2)^2\tag{1} $$ Thus, the centers will lie on the line $$ 2(x_2-x_1)x+2(y_2-y_1)y+(x_1^2-x_2^2+y_1^2-y_2^2)=0\tag{2} $$ Now, we can use $(2)$ to solve for $y$ in terms of $x$ and substitute that into $$ (x-x_1)^2+(y-y_1)^2=r^2\tag{3} $$ and solve the quadratic equation for $x$, then plug in to $(2)$ to get $y$.

robjohn
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