# Apollonian Circles Theorem

Given two points $A$ and $B$ and a number $r.$ What is the locus of points $P$ such that $AP/BP = r?$ The answer is a circle. The circle which is known as the Apollonian Circle. For every positive r there is a different one.

This problem has been treated elsewhere. Here we present a different solution based on the inversion transform. Along the way we show that the whole family of Apollonian circles can be inverted into a family of concentric circles.

Solution ### Apollonian Circles Theorem

For two fixed points $A$ and $B$ and a real $r\gt 0,$ the locus of points $P$ such that $AP/BP = r$ is a circle. The circle is known as the Apollonian Circle. Consider a circle of radius $R$ centered at $A.$ Let $t$ be the inversion in this circle. Set $t(B) = B'$ and $t(P) = P'.$ The calculations are simplified by taking $R = 1,$ but, in principle, any circle with center at $A$ will do. So, assume $R = 1.$

This means, in particular, that

(1)

$AP\cdot AP' = AB\cdot AB' = 1.$

Triangles $AB'P'$ and $APB,$ in which the sides satisfy (1), also share the angle at $A.$ They are, therefore, similar. It then follows that

(2)

$BP / B'P' = AB / AP'.$

From here,

(3)

$B'P' = BP\cdot AP' / AB.$

Combining (1) and (3) gives

(4)

\begin{align} B'P' &= BP / (AP\cdot AB)\\ &= 1/rAB. \end{align}

This tells us that $P'$ lies on a circle of radius $1/rAB$ centered at $B'.$ In other words, the inversive image of the locus of points $P$ is a circle centered at $B'.$ To repeat, the Apollonian circle defined by the point circles $A$ and $B$ and $r \gt 0$ is mapped by $t$ to a circle centered at $t(B).$ This is true for any Apollonian circle defined by $A$ and $B,$ so that the whole family of them is mapped on the family of concentric circles with center $t(B).$

In establishing Steiner's Porism we showed that any two non-intersecting circles can be inverted into a pair of concentric circles. The above strengthens this assertion with a more direct proof.

The Apollonian circles defined by two point circles are said to be coaxal. This is one of the three varieties of coaxal families.

### References

1. D. A. Brannan et al, Geometry, Cambridge University Press, 2002 ### Inversion - Introduction 