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In Geometry, a locus (plural: loci) is a set of points whose location satisfies or is determined by one or more specified conditions i.e., 1)every point satisfies a given condition and 2)every point satisfiying it is in that particular locus
Commonly studied loci
Examples from plane geometry:
- The set of points equidistant from two points is a perpendicular bisector to the line segment connecting the two points.
- The set of points equidistant from two lines which cross is the angle bisector.
- All conic sections are loci:
- Parabola: the set of points equidistant from a single point (the focus) and a line (the directrix).
- Circle: the set of points for which the distance from a single point is constant (the radius). The set of points for each of which the ratio of the distances to two given foci is a positive constant (that is not 1) is referred to as a Circle of Apollonius.
- Hyperbola: the set of points for each of which the absolute value of the difference between the distances to two given foci is a constant.
- Ellipse: the set of points for each of which the sum of the distances to two given foci is a constant. In particular, the circle is a locus.
Proof of a Locus
In order to prove the correctness of a locus, one generally divides the proof into two stages:
- Proof that all the points that satisfy the conditions are on the locus.
- Proof that all the points on the locus satisfy the conditions.
This is equivalent to
This equation represents a circle.
- Robert Clarke James, Glenn James: Mathematics Dictionary. Springer 1992, ISBN 978-0-412-99041-0, p. 255 (restricted online copy, p. 255, at Google Books)
- Alfred North Whitehead: An Introduction to Mathematics. BiblioBazaar LLC 2009 (reprint), ISBN 978-1-103-19784-2, pp. 121 (restricted online copy, p. 121, at Google Books)
- George Wentworth: Junior High School Mathematics: Book III. BiblioBazaar LLC 2009 (reprint), ISBN 978-1-103-15236-0, pp. 265 (restricted online copy, p. 265, at Google Books)