Circle of Apollonius is Circle/Proof 2

Proof
Let $A = \tuple {x_a, y_a}, B = \tuple {x_b, y_b}$.

Let $X = \tuple {x, y}$.

We have:

From Equation of Circle in Cartesian Plane, this is an equation in the form:


 * $A \paren {x^2 + y^2} + B x + C y + D = 0$

with radius $R$ and center $\tuple {a, b}$, where:
 * $R = \dfrac 1 {2 A} \sqrt {B^2 + C^2 - 4 A D}$
 * $\tuple {a, b} = \tuple {\dfrac B {2 A}, \dfrac C {2 A} }$

Here, we have:

Hence the center $\tuple {a, b}$ can be evaluated:

Then the radius can be evaluated: