Equidistance

Equidistance is a concept in geometry describing the state of being at the same distance from two or more reference objects, using a given metric—most often the standard Euclidean distance. In many geometric problems, equidistance defines a locus: the set of all points that satisfy equality of distances to the reference objects.

The classic case is two fixed points A and B. The locus of points equidistant from A and B is the perpendicular bisector of the segment AB. In the plane this is a straight line; in space it is a plane. This line or plane consists of all points whose distances to A and B are equal.

If the references are two lines l1 and l2, the locus of points equidistant from the lines is the pair of angle bisectors of the lines (the internal and external bisectors). When the references are a point F and a line d, the locus is a parabola: each point on the parabola is as far from F as from d, with F as the focus and d as the directrix.

Other common cases include three non-collinear points A, B, and C, where the point equidistant from all three—if it exists in the plane—is the circumcenter, found at the intersection of the pairwise perpendicular bisectors of the sides. In higher dimensions, analogous loci arise, such as equidistant planes in three-space.

Equidistance concepts underlie constructions, surveying, and computational geometry (for example, Voronoi diagrams rely on equidistance to seed points). The term derives from equi- meaning equal and distance, reflecting its central idea.