A Special Theorem: Part 2 (V1)

In the applet below, the perpendicular bisector of the blue segment (with endpoints A and B) is shown.   Before completing the directions below, move/drag points A and B around to verify that the brown line is the perpendicular bisector of AB.     Directions: Use the tools of GeoGebra to do the following: 1) Plot a point anywhere on this perpendicular bisector.   2) Measure and display the distance from this point to point A. 3) Measure and display the distance from this point to point B. 3) Now drag this point along the perpendicular bisector as much as you'd like.     Be sure to zoom out and keep dragging this point along this perpendicular bisector.     What do you notice? 
4) Use your observations to complete the following statement:       If a point lies on the ______________________    ___________________ of a      ________________________, then that ____________ is ___________________    from the ____________________ of that _____________________.   5) Prove the statement (you completed in step (4) above) using the format of a 2 column proof.