circle of Apollonius

== English == === Etymology === Named for the ancient Greek geometer and astronomer Apollonius of Perga (ca 262—ca 190 BCE). === Noun === circle of Apollonius (plural circles of Apollonius) (geometry) The figure (a generalised circle) definable as the locus of points P such that, for given points A, B and C, | A P | | B P | = | A C | | B C | {\displaystyle {\frac {|AP|}{|BP|}}={\frac {|AC|}{|BC|}}} . (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). ==== Usage notes ==== (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. === See also === Apollonian circles on Wikipedia.Wikipedia Apollonius pursuit problem on Wikipedia.Wikipedia Circle on Wikipedia.Wikipedia Problem of Apollonius on Wikipedia.Wikipedia