Apollonius Point
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In
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, the Apollonius point is a
triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For exampl ...
designated as ''X''(181) in
Clark Kimberling Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer seq ...
's
Encyclopedia of Triangle Centers The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or " centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville. , the ...
(ETC). It is defined as the point of
concurrence In Western jurisprudence, concurrence (also contemporaneity or simultaneity) is the apparent need to prove the simultaneous occurrence of both ("guilty action") and ("guilty mind"), to constitute a crime; except in crimes of strict liability ...
of the three line segments joining each vertex of the triangle to the points of
tangency In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
formed by the opposing
excircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
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 point In Euclidean geometry, the isodynamic points of a triangle are points associated with the triangle, with the properties that an inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the dis ...
s of a triangle. This usage could also be justified on the ground that the isodynamic points are related to the three
Apollonian circles In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. Th ...
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 be any given triangle. Let the
excircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
s of opposite to the vertices be respectively. Let be the circle which touches the three excircles such that the three excircles are within . Let be the points of contact of the circle with the three excircles. The lines are
concurrent Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to: Law * Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea'' * Concurring opinion (also called a "concurrence"), a ...
. The point of concurrence is the ''Apollonius point'' of . 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 referred to in the above definition is one of these eight circles touching the three excircles of triangle . In
Encyclopedia of Triangle Centers The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or " centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville. , the ...
the circle is the called the ''Apollonius circle'' of .


Trilinear coordinates

The trilinear coordinates of the Apollonius point are :\frac : \frac : \frac :=\sin^2 A \cos^2 \left(\frac-\frac\right): \sin^2B\cos^2 \left(\frac-\frac\right) : \sin^2C\cos^2 \left(\frac-\frac\right).


References

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See also

*
Apollonius' theorem In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that "the sum of the squares of any two sides of any triangle equals twice the square on half the third side, t ...
* Apollonius of Perga (262–190 BC), geometer and astronomer * Apollonius problem *
Apollonian circles In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. Th ...
*
Isodynamic point In Euclidean geometry, the isodynamic points of a triangle are points associated with the triangle, with the properties that an inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the dis ...
of a triangle Triangle centers