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In triangle geometry, the 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.
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In the literature, the term "Apollonius points" has also been used to refer to the isodynamic points of a triangle. 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.
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 EA, EB, EC respectively. Let E be the circle which touches the three excircles EA, EB, EC 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.
Trilinear coordinates
The trilinear coordinates of the Apollonius point are