The bisector of a fixed point p and a smooth plane curve C-i.e., the locus traced by a point that remains equidistant with respect to p and C-is investigated in the case that C admits a regular polynomial or rational parameterization. It is shown that the bisector may be regarded as (a subset of) a "variable-distance" offset curve to C which has the attractive property, unlike fixed-distance offsets, of being generically a rational curve. This "untrimmed bisector" usually exhibits irregular points and self-intersections similar in nature to those seen on fixed-distance offsets. A trimming procedure, which identifies the parametric subsegments of this curve that constitute the true bisector, is described in detail. The bisector of the point p and any finite segment of the curve C is also discussed. © 1994.