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Apollonius Circle

There are four completely different definitions of the so-called Apollonius circles:

1. The set of all points whose distances from two fixed points are in a constant ratio (Durell 1928, Ogilvy 1990).

2. One of the eight circles that is simultaneously tangent to three given circles (i.e., a circle solving Apollonius' problem for three circles).

3. One of the three circles passing through a vertex and both isodynamic points and of a triangle (Kimberling 1998, p. 68).

4. The circle that touches all three excircles of a triangle and encompasses them (Kimberling 1998, p. 102).

Given one side of a triangle and the ratio of the lengths of the other two sides, the locus of the third polygon vertex is the Apollonius circle (of the first type) whose center is on the extension of the given side. For a given triangle, there are three circles of Apollonius. Denote the three Apollonius circles (of the first type) of a triangle by , , and , and their centers , , and . The center is the intersection of the side with the tangent to the circumcircle at . is also the pole of the symmedian point with respect to circumcircle. The centers , , and are collinear on the polar of with regard to its circumcircle, called the Lemoine axis. The circle of Apollonius is also the locus of a point whose pedal triangle is isosceles such that .

The eight Apollonius circles of the second type are illustrated above.

Let and be points on the side line of a triangle met by the interior and exterior angle bisectors of angles . Then the circle with diameter is called the -Apollonian circle. Similarly, construct the - and -Apollonian circles (Johnson 1929, pp. 294-299). The Apollonian circles pass through the vertices , , and , and through the two isodynamic points and (Kimberling 1998, p. 68). The -Apollonius circle has center with trilinears

 (1)

 (2)

where is the circumradius of the reference triangle.

Because the Apollonius circles intersect pairwise in the isodynamic points, they share a common radical line

 (3)

which is the central line corresponding to Kimberling center , the isogonal conjugate of the Kiepert parabola focus .

The vertices of the D-triangle lie on the respective Apollonius circles.

The circle which touches all three excircles of a triangle and encompasses them is often known as "the" Apollonius circle (Kimberling 1998, p. 102). It has circle function

 (4)

which corresponds to Kimberling center . Its center has triangle center function

 (5)

which is Kimberling center . Its radius is

 (6)

where is the inradius and is the semiperimeter of the reference triangle. It can be constructed as the inversive image of the nine-point circle with respect to the circle orthogonal to the excircles of the reference triangle. It is a Tucker circle (Grinberg and Yiu 2002).

Kimberling centers for , 2038, 3029, 3030, 3031, 3032, 3033, and 3034 lie on the Apollonius circle. It is also orthogonal to the Stevanović circle.

Apollonian Gasket, Apollonius Point, Apollonius' Problem, Apollonius Pursuit Problem, Casey's Theorem, D-Triangle, Hart's Theorem, Hexlet, Isodynamic Points, Soddy Circles, Tangent Circles, Tangent Spheres

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References

Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 16, 1928.Gallatly, W. "The Apollonian Circles." §127 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 92, 1913.Grinberg, D. and Yiu, P. "The Apollonius Circle as a Tucker Circle." Forum Geom. 2, 175-182, 2002. http://forumgeom.fau.edu/FG2002volume2/FG200222index.html.Herrmann, M. "Eine Verallgemeinerung des Apollonischen Problems." Math. Ann. 145, 256-264, 1962.Kasner, E. and Supnick, F. "The Apollonian Packing of Circles." Proc. Nat. Acad. Sci. USA 29, 378-384, 1943.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 40 and 294-299, 1929.Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 14-23, 1990.Trott, M. Graphica 1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael Trott. Champaign, IL: Wolfram Media, pp. 60 and 88, 1999.

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Apollonius Circle

Cite this as:

Weisstein, Eric W. "Apollonius Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ApolloniusCircle.html