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Motivated by the work of Asano et al.[l], we consider the distance trisector problem and Zone diagram considering segments in the plane as the input geometric objects. As the most basic case, we first consider the pair of curves (distance trisector curves) trisecting the distance between a point and a line. This is a natural extension of the bisector curve (that is a parabola) of a point and a line. In this paper, we show that these trisector curves C1 and C2 exist and are unique. We then give a practical algorithm for computing the Zone diagram of a set of segments in a digital plane.