In the literature, the term "Apollonius points" has also been used to refer to the isodynamic points of a triangle.^{[1]} This usage could also be justified on the ground that the isodynamic points are related to the three Apollonian circles associated with a triangle.

The solution of the Apollonius problem has been known for centuries. But the Apollonius point was first noted in 1987.^{[2]}^{[3]}

Definition

The Apollonius point of a triangle is defined as follows.

Let △ABC be any given triangle. Let the excircles of △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 △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 △ABC.

Trilinear coordinates

The trilinear coordinates of the Apollonius point are^{[2]}

^Katarzyna Wilczek (2010). "The harmonic center of a trilateral and the Apollonius point of a triangle". Journal of Mathematics and Applications. 32: 95–101.