Editing Equidistant (section)

{{Short description|Concept in geometry}}
{{redirect|Equidistance|the principle in maritime boundary claims|Equidistance principle}}
{{more citations needed|date=August 2012}}

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[[File:Perpendicular bisector.gif|right|thumb|[[Perpendicular bisector]] of a line segment. The point where the red line crosses the black line segment is equidistant from the two end points of the black line segment.]]
[[File:Circumscribed Polygon.svg|thumb|The [[cyclic polygon]] P is [[circumscribed circle|circumscribed]] by the circle C. The circumcentre O is equidistant to each point on the circle, and a fortiori to each vertex of the polygon.]]

A point is said to be '''equidistant''' from a set of objects if the [[distance]]s between that point and each object in the set are equal.<ref>{{cite book
 |title=The concise Oxford dictionary of mathematics
 |first1= Christopher |last1=Clapham
 |first2= James |last2=Nicholson
 |publisher=Oxford University Press
 |isbn=978-0-19-923594-0
 |year=2009
 |pages=164–165
 |url=https://books.google.com/books?id=UTCenrlmVW4C&pg=PT164
}}</ref>

In two-dimensional [[Euclidean geometry]], the [[locus (mathematics)|locus]] of points equidistant from two given (different) points is their [[perpendicular bisector]]. In three dimensions, the locus of points equidistant from two given points is a plane, and generalising further, in [[n-dimensional space]] the locus of points equidistant from two points in ''n''-space is an (''n''&minus;1)-space.

For a [[triangle]] the [[circumcentre]] is a point equidistant from each of the three [[vertex (geometry)|vertices]]. Every non-degenerate triangle has such a point. This result can be generalised to [[cyclic polygon]]s: the circumcentre is equidistant from each of the vertices. Likewise, the [[incenter|incentre]] of a triangle or any other [[tangential polygon]] is equidistant from the points of tangency of the polygon's sides with the circle. Every point on a [[Bisection#Bisectors of the sides of a polygon|perpendicular bisector of the side]] of a triangle or other polygon is equidistant from the two vertices at the ends of that side. Every point on the [[Bisection#Angle bisector|bisector of an angle]] of any polygon is equidistant from the two sides that emanate from that angle.

The center of a [[rectangle]] is equidistant from all four vertices, and it is equidistant from two opposite sides and also equidistant from the other two opposite sides. A point on the [[axis of symmetry]] of a [[kite (geometry)|kite]] is equidistant between two sides.

The center of a [[circle]] is equidistant from every point on the circle. Likewise the center of a [[sphere]] is equidistant from every point on the sphere.

A [[parabola]] is the set of points in a plane equidistant from a fixed point (the [[Focus (geometry)|focus]]) and a fixed line (the directrix), where distance from the directrix is measured along a line perpendicular to the directrix.

In [[shape analysis (digital geometry)|shape analysis]], the [[topological skeleton]] or [[medial axis]] of a [[shape]] is a thin version of that shape that is equidistant from its [[boundary (topology)|boundaries]].

In [[Euclidean geometry]], [[parallel lines]] (lines that never intersect) are equidistant in the sense that the distance of any point on one line from the nearest point on the other line is the same for all points.

In [[hyperbolic geometry]] the set of points that are equidistant from and on one side of a given line form a [[hypercycle (hyperbolic geometry)|hypercycle]] (which is a curve, not a line).<ref>{{citation|first=James R.|last=Smart|title=Modern Geometries|edition=5th|publisher=Brooks/Cole|year=1997|isbn=0-534-35188-3|page=392}}</ref>