Equidistant

For a triangle the circumcentre is a point equidistant from each of the three vertices.

This result can be generalised to cyclic polygons: the circumcentre is equidistant from each of the vertices.

A point on the axis of symmetry of a kite is equidistant between two sides.

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 hyperbolic geometry the set of points that are equidistant from and on one side of a given line form a hypercycle (which is a curve, not a line).

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.
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.