circle of Apollonius

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

Noun

 * 1)  The figure (a generalised circle) definable as the locus of points P such that, for given points A, B and C, $$\frac{|AP|}{|BP|} = \frac{|AC|}{|BC|}$$.
 * 2)  Any of eight circles (including degenerate cases) that, for a given set of three circles, solve the  (i.e., intersect each circle tangentially).
 * 1)  Any of eight circles (including degenerate cases) that, for a given set of three circles, solve the  (i.e., intersect each circle tangentially).
 * 1)  Any of eight circles (including degenerate cases) that, for a given set of three circles, solve the  (i.e., intersect each circle tangentially).
 * 1)  Any of eight circles (including degenerate cases) that, for a given set of three circles, solve the  (i.e., intersect each circle tangentially).

Usage notes
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.

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.