Circle of Apollonius is Circle

Theorem
Let $A, B$ be distinct points in the plane.

Let $\lambda \in \R_{>0}$ be a strictly positive real number.

Let $X$ be the locus of points in the plane such that:
 * $XA = \lambda \paren {XB}$

Then $X$ is in the form of a circle, known as a circle of Apollonius.


 * Circle-of-Apollonius.png

If $\lambda < 1$, then $A$ is inside the circle, and $B$ is outside.

If $\lambda > 1$, then $B$ is inside the circle, and $A$ is outside.