A Special Theorem: Part 2 (V3)

Interact with the applet below for a few minutes. Then answer the questions that follow. Be sure to alter the locations of points A, B, and C each time before (and even after) you re-slide the slider!
Questions: 1) How would you describe the blue line p with respect to the segment with endpoints A and B? What exactly is line p? (Hint: It is a special vocabulary term we've already learned! If you're still stumped, consider what the measure of the gray angle. What is it?) 2) Notice how point C always stays on line p. What can you conclude about the distances AC and BC? 3) Formally prove, (in the format of a 2-column proof, paragraph proof, or coordinate-geometry proof), that your conclusion for (2) is true for any pointC that lies on line p.