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Bisection of two circular segments by a line. All the points on the red line are equidistant from the two end points of the black line.
A shape and its skeleton, computed with a topology-preserving thinning algorithm.
The cyclic polygon P is 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 distances between that point and each object in the set are equal.[1]

In two-dimensional Euclidian geometry the 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−1)-space.

For a triangle the circumcentre is a point equidistant from each of the three end points. Every non degenerate triangle has such a point. This result can be generalised to cyclic polygons. 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) and a fixed line (the directrix), where distance from the directrix is measured along a line perpendicular to the directrix.

In shape analysis, the topological skeleton or medial axis of a shape is a thin version of that shape that is equidistant from its boundaries.


  1. ^ Clapham, Christopher; Nicholson, James (2009). The concise Oxford dictionary of mathematics. Oxford University Press. pp. 164–165. ISBN 978-0-19-923594-0.