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Perpendicular Bisector (Becker-Revised)

This worksheet is a modification of a worksheet created by Tim Brzezinski.

In the applet below, line p is the perpendicular bisector of the segment with endpoints A and B. The slider on the right gives insight into a theorem that holds true for every point that lies on the perpendicular bisector of a segment. Interact with this applet for a few minutes. As you do, be sure to change the location of the white point C each time before you re-slide the slider.
Questions: 1) What do you notice about the distances (lengths) AC and BC? 2) Does your answer to question (1) above hold true for every point on this perpendicular bisector? That is, is your response to question (1) the same regardless of where point C lies? 3) Let D be the point where segment AB intersects line p. Using similar triangles, which triangle cases proves that triangle ACD is congruent to triangle BCD? 4) Prove this assertion (2 and 3) true in the format of a 2-column proof.