A Special Theorem: Part 1 (V1)

In the applet below, note that point C is equidistant from A and B.   In this applet, C will ALWAYS REMAIN EQUIDISTANT from A and B.  Also note that A and B serve as endpoints of a segment.   Directions: 1) Drag C around as much as you'd like (without moving A and B).     What can you conclude about the locus (set of points) in the plane     that are equidistant from the endpoints of a segment?     What does this locus look like?   2) Let's test this conjecture again.      Change the location of point A and point B.       Hit the "Clear Trace" button to erase the previous traces of point C.       Repeat Step 1.   3) Use the tools of GeoGebra to now show that your conjecture is true.