Editing Circle (section)

==Circle of Apollonius==
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{{see also|Circles of Apollonius}}
[[Image:Apollonius circle definition labels.svg|thumb|250px|left|Apollonius' definition of a circle: {{nowrap|''d''<sub>1</sub>/''d''<sub>2</sub>}} constant]]
[[Apollonius of Perga]] showed that a circle may also be defined as the set of points in a plane having a constant ''ratio'' (other than 1) of distances to two fixed foci, ''A'' and ''B''.<ref>{{cite journal | last = Harkness | first = James | title = Introduction to the theory of analytic functions |journal=Nature |volume=59 |issue=1530 |year=1898 |page = 30 | url = http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=01680002 |archive-url=https://web.archive.org/web/20081007134238/http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=01680002 | url-status = dead |archive-date=2008-10-07 |bibcode=1899Natur..59..386B |doi=10.1038/059386a0 |s2cid=4030420 }}</ref><ref>[[C. Stanley Ogilvy|Ogilvy, C. Stanley]], ''Excursions in Geometry'', Dover, 1969, 14–17.</ref> (The set of points where the distances are equal is the perpendicular bisector of segment ''AB'', a line.) That circle is sometimes said to be drawn ''about'' two points.

The proof is in two parts. First, one must prove that, given two foci ''A'' and ''B'' and a ratio of distances, any point ''P'' satisfying the ratio of distances must fall on a particular circle. Let ''C'' be another point, also satisfying the ratio and lying on segment ''AB''. By the [[angle bisector theorem]] the line segment ''PC'' will bisect the [[interior angle]] ''APB'', since the segments are similar:
<math display="block">\frac{AP}{BP} = \frac{AC}{BC}.</math>

Analogously, a line segment ''PD'' through some point ''D'' on ''AB'' extended bisects the corresponding exterior angle ''BPQ'' where ''Q'' is on ''AP'' extended. Since the interior and exterior angles sum to 180 degrees, the angle ''CPD'' is exactly 90 degrees; that is, a right angle. The set of points ''P'' such that angle ''CPD'' is a right angle forms a circle, of which ''CD'' is a diameter.

Second, see<ref>Altshiller-Court, Nathan, ''College Geometry'', Dover, 2007 (orig. 1952).</ref>{{rp|p=15}} for a proof that every point on the indicated circle satisfies the given ratio.

===Cross-ratios===
A closely related property of circles involves the geometry of the [[cross-ratio]] of points in the complex plane. If ''A'', ''B'', and ''C'' are as above, then the circle of Apollonius for these three points is the collection of points ''P'' for which the absolute value of the cross-ratio is equal to one:
<math display="block">\bigl|[A, B; C, P]\bigr| = 1.</math>

Stated another way, ''P'' is a point on the circle of Apollonius if and only if the cross-ratio {{nobr|[''A'', ''B''; ''C'', ''P'']}} is on the unit circle in the complex plane.

==={{anchor|Generalized circles}} Generalised circles===
{{See also|Generalised circle}}
If ''C'' is the midpoint of the segment ''AB'', then the collection of points ''P'' satisfying the Apollonius condition
<math display="block">\frac{|AP|}{|BP|} = \frac{|AC|}{|BC|}</math>
is not a circle, but rather a line.

Thus, if ''A'', ''B'', and ''C'' are given distinct points in the plane, then the [[Locus (mathematics)|locus]] of points ''P'' satisfying the above equation is called a "generalised circle." It may either be a true circle or a line. In this sense a line is a generalised circle of infinite radius.