Apollonius point

In Euclidean geometry, the Apollonius point is a triangle center designated as X(181) in Clark Kimberling's Encyclopedia of Triangle Centers (ETC). It is defined as the point of concurrence of the three line segments joining each vertex of the triangle to the points of tangency formed by the opposing excircle and a larger circle that is tangent to all three excircles.

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

Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line

The Apollonius point of a triangle is defined as follows.

Let Template:Math be any given triangle. Let the excircles of Template:Math opposite to the vertices Template:Mvar be Template:Mvar respectively. Let Template:Mvar be the circle which touches the three excircles Template:Mvar such that the three excircles are within Template:Mvar. Let Template:Mvar be the points of contact of the circle Template:Mvar with the three excircles. The lines Template:Mvar are concurrent. The point of concurrence is the Apollonius point of Template:Math.

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 Template:Mvar referred to in the above definition is one of these eight circles touching the three excircles of triangle Template:Math. In Encyclopedia of Triangle Centers the circle Template:Mvar is the called the Apollonius circle of Template:Math.

Trilinear coordinates

The trilinear coordinates of the Apollonius point are

$\begin{array}{cccccc}& \frac{a(b+c{)}^2}{b+c-a}& :& \frac{b(c+a{)}^2}{c+a-b}& :& \frac{c(a+b{)}^2}{a+b-c}\\ =& {\sin }^{2}A{\cos }^2\frac{B-C}{2}& :& {\sin }^{2}B{\cos }^2\frac{C-A}{2}& :& {\sin }^{2}C{\cos }^2\frac{A-B}{2}\end{array}$

See also

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