Locus of Points Equidistant Between Two Points
Intuitively the set of points equidistant from two given points is the set of points halfway between them. The locus will be a straight line halfway between the two points. We can construct this line by drawing circles of equal radius centred at A and B. The radius should be over half the distance AB. These two circles will intersect at two points. If you draw a line between the points where the circles cross, this line will be the required locus.
For example, to find the locus of points equidistant from A and B below.
Draw an arc of a circle centred at A as below.
Now move the compass needle to B and draw an arc of a circle. Make sure that the arcs are big enough to obtain two crossing points.
Between the points where the arcs cross, draw a line.
This line will be the required locus of points.