circle of Apollonius

plural circles of Apollonius

1. geometryThe figure (a generalised circle) definable as the locus of points P such that, for given points A, B and C, =/(\frac).

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

   1934, Tôhoku Mathematical Journal, volumes 39-40, page 264

   2 more examples

   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.

   1995, Howard Whitley Eves, College Geometry, page 165

   More generally, the locus of points z such that #92;left#92;vertx-z#92;right#92;vert#61;#92;alpha#92;left#92;verty-z#92;right#92;vert is a circle of Apollonius which has x and y as inverse points and which cuts each circle through x and 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 #92;mathcal#123;l#125;#95;p-norm.

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

2. geometryAny 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).