Apollonius point

In triangle geometry, Apollonius point is a special point associated with a plane triangle. The point is a triangle center and it is designated as X(181) in Clark Kimberling's Encyclopedia of Triangle Centers (ETC). The Apollonius center is also related to the Apollonius problem.

In the literature, the term "Apollonius points" has also been used to refer to those points which are called as the isodynamic points of a triangle in the Encyclopedia of Triangle Centers (the points designated as X(15) and X(16)). This usage could also be justified on the ground that the isodynamic points in the sense of ETC 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.

Definition

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 tngent 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 excirxles of triangle ABC. In Encyclopedia of Triangle Centers the circle E is the called the Apolonius circle of triangle ABC.

Trilinear coordinates

The trilinear coordiantes of the Apollonius point:

( a ( b + c )² / ( b + c − a ) : b ( c + a )² / ( c + a − b ) : c ( a + b )² / ( a + b − c )

= ( sin A cos ( B/2 − C/2 ) )² : ( sin B cos (C/2 − A/2) )² : ( sin C cos (A/2 − B/2) )²