Apollonius point

In

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

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]

${\begin{array}{ccccc}&\displaystyle {\frac {a(b+c)^{2}}{b+c-a}}&:&\displaystyle {\frac {b(c+a)^{2}}{c+a-b}}&:&\displaystyle {\frac {c(a+b)^{2}}{a+b-c}}\\[4pt]=&\sin ^{2}\!A\,\cos ^{2}{\frac {B-C}{2}}&:&\sin ^{2}\!B\,\cos ^{2}{\frac {C-A}{2}}&:&\sin ^{2}\!C\,\cos ^{2}{\frac {A-B}{2}}\end{array}}$

References

^ Katarzyna Wilczek (2010). "The harmonic center of a trilateral and the Apollonius point of a triangle". Journal of Mathematics and Applications. 32: 95–101.
^ ^a ^b Kimberling, Clark. "Apollonius Point". Archived from the original on 10 May 2012. Retrieved 16 May 2012.
^ C. Kimberling; Shiko Iwata; Hidetosi Fukagawa (1987). "Problem 1091 and Solution". Crux Mathematicorum. 13: 217–218.

Definition

Trilinear coordinates

References

See also