Draw two intersecting lines l and m intersecting at O. Draw OA the bisector of 
. Mark E on OA. Draw EB and EC perpendicular to OB and OC respectively. EB and EC are the distances of E from OB and OC respectively.
EB=EC
Similarly take some more points on the bisector of. From each point on OA draw perpendiculars to OB and OC. You will find that the perpendicular distance from any point on the bisector of on OB and OC always to be equal. This means the locus of a point equidistant from two intersecting lines is the union of the pair of lines bisecting the angles formed by the given lines. 