Apollonius point

The Apollonius point of a triangle is defined as follows.

Let ABC be any given triangle. Let the excircles of triangle ABC opposite to the vertices A, B, C be E_A, E_B, E_C respectively. Let E be the circle which touches the three excircles E_A, E_B, E_C such that the three excircles are within E. Let A' , B' , C' be the points of contact of the circle E with the three excircles. The lines AA' , BB' , CC' are concurrent. The point of concurrence is the Apollonius point of triangle ABC.

The Apollonius problem is the problem of constructing a circle tangent to three given circles in a plane. In general, there are eight circles touching three given circles. The circle E referred to in the above definition is one of these eight circles touching the three excircles of triangle ABC. In Encyclopedia of Triangle Centers the circle E is the called the Apollonius circle of triangle ABC.

The trilinear coordinates of the Apollonius point are[2]

${\displaystyle {\frac {a(b+c)^{2}}{b+c-a}}:{\frac {b(c+a)^{2}}{c+a-b}}:{\frac {c(a+b)^{2}}{a+b-c}}}$

${\displaystyle =\sin ^{2}A\cos ^{2}({\frac {B}{2}}-{\frac {C}{2}}):\sin ^{2}B\cos ^{2}({\frac {C}{2}}-{\frac {A}{2}}):\sin ^{2}C\cos ^{2}({\frac {A}{2}}-{\frac {B}{2}}).}$

• Apollonius' theorem
• Apollonius of Perga (262–190 BC), geometer and astronomer
• Apollonius problem
• Apollonian circles
• Isodynamic point of a triangle