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 1:mu (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 S and S^' 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 k_1, k_2, and k_3, and their centers L_1, L_2, and L_3. The center L_1 is the intersection of the side A_2A_3 with the tangent to the circumcircle at A_1. L_1 is also the pole of the symmedian point K with respect to circumcircle. The centers L_1, L_2, and L_3 are collinear on the polar of K with regard to its circumcircle, called the Lemoine axis. The circle of Apollonius k_1 is also the locus of a point whose pedal triangle is isosceles such that P_1P_2^_=P_1P_3^_.


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


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


and radius


where R is the circumradius of the reference triangle.


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


which is the central line L_(523) corresponding to Kimberling center X_(523), the isogonal conjugate of the Kiepert parabola focus X_(110).

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


which corresponds to Kimberling center X_(940). Its center has triangle center function


which is Kimberling center X_(970). Its radius is


where r is the inradius and s 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 X_i for i=2037, 2038, 3029, 3030, 3031, 3032, 3033, and 3034 lie on the Apollonius circle. It is also orthogonal to the Stevanović circle.

See also

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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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., 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.

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