Apollonius's circle

Apollonius's circle is a special circle related to a triangle. Given a triangle $ABC$ and a ratio $k > 0$, the locus of points $P$ such that $\frac{PB}{PC} = k$ is a circle (unless $k=1$, in which case the locus is the angle bisector). This circle is called an Apollonius circle of triangle $ABC$ with respect to $B$ and $C$.

Important properties

• For any two distinct points $B$ and $C$ and any positive real number $k \neq 1$, the set of points $P$ such that $\frac{PB}{PC} = k$ forms a circle.

• If $k = 1$, the locus is the perpendicular bisector of $BC$.

• The center and radius of the Apollonius circle can be constructed using compass and straightedge.

• Apollonius's circles are useful in triangle geometry, especially in problems involving ratios of distances.