Being at an equal distance from the same point or thing.
In the equidistant projections scale is preserved (not distorted) in the direction perpendicular to the line of zero distortion or radially outwards from a point of zero distortion. The name arises from the fact that in the normal aspect of Cylindrical, Conic and Azimuthal projections the principal scale is preserved along the meridians and therefore all parallels on the map are equidistantly spaced.
The same distance.
A map projection is equidistant if it maintains constant scale and true distance only from the centre of the projection or along great circles (meridians) passing through this point. In other words, a planar equidistant projection centred on Toronto would show the correct distance to any other location on the map, from Toronto only. This property is achieved at the expense of distorting area and direction.
Equidistant maps show true distances only from the center of the projection or along a special set of lines. For example, an Azimuthal Equidistant map centered at Washington shows the correct distance between Washington and any other point on the projection. It shows the correct distance between Washington and San Diego and between Washington and Seattle. But it does not show the correct distance between San Diego and Seattle. No flat map can be both equidistant and equal area.
Applied to maps which have selected lines along which distances, or scale, can be measured correctly.
Having equal distances.