circle of Apollonius

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Circles of Apollonius

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Named for the ancient Greek geometer and astronomer Apollonius of Perga (ca 262—ca 190 BCE).

circle of Apollonius (plural circles of Apollonius)

1. (geometry) The figure (a generalised circle) definable as the locus of points P such that, for given points A, B and C, ${\displaystyle {\frac {|AP|}{|BP|}}={\frac {|AC|}{|BC|}}}$.
   • 1934, Tôhoku Mathematical Journal, volumes 39-40, page 264:

     In the present paper an attempt is made to find for the tetrahedron the analogues of the circles of Apollonius of the triangle.

   • 1995, Howard Whitley Eves, College Geometry‎, page 165:

     It follows that O lies on the circle of Apollonius for A and C and ratio OT/OC, and on the circle of Apollonius for B and C and ratio OT/OC. Point O is thus found at the intersections, if any exist, of these two circles of Apollonius. The details are left to the reader.

   • 1996, A. C. Thompson, Minkowski Geometry‎, page 126:

     More generally, the locus of points ${\displaystyle z}$ such that ${\displaystyle \left\vert x-z\right\vert =\alpha \left\vert y-z\right\vert }$ is a circle of Apollonius which has ${\displaystyle x}$ and ${\displaystyle y}$ as inverse points and which cuts each circle through ${\displaystyle x}$ and ${\displaystyle y}$ orthogonally. […] They show loci analogous to circles of Apollonius and the locus of points equidistant from two given points when the norm is an ${\displaystyle {\mathcal {l}}_{p}}$-norm.

2. (geometry) Any of eight circles (including degenerate cases) that, for a given set of three circles, solve the problem of Apollonius (i.e., intersect each circle tangentially).

(generalised circle): In explanations of the construction, C is sometimes shown as collinearly between A and B, but this is merely a convenience of explanation. The figure will, however, always intersect the segment at a single point. In most cases the locus of P is a circle, but in the case that C is the midpoint of AB, the result is the line perpendicular to the segment at C, thus justifying the use of the term generalised circle (“a circle or a line”).

The three circles of Apollonius of a triangle are the three such figures obtainable by letting AB be one of the sides of the triangle and C be the vertex opposite.

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