Find the centers of the two possible circles formed given two points on their circumference.

Question

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?

Answer 1

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 $y$.