Another Special Theorem: Part 2 (V1)

In the applet below, the angle bisector of the ANGLE BAC is shown.   Point E is a point that lies on this angle bisector.  (Feel free to drag it around.) Before completing the directions below, move/drag points B, A, and/or C around to verify that the pink ray still remains an angle bisector of ANGLE BAC.     Directions:  1) Use the tools of GeoGebra to measure the distance from E to each side (ray) of     ANGLE BAC.     (Note: It should be obvious to you that this is not the same as finding EB and EC.      Think about what you need to do.)      2) What do you notice?   3) Now move point E along this angle bisector.  Does your observation in (2) still hold true? 4) Now move/drag points BA, and/or C around. Does your observation in (2) still hold true?     
5) Use your observations above to complete the following statement:       If a ______________ lies on the ________________ of an _______________, then that     ______________ is _____________________ from the __________ of that ___________.   6) Now prove this statement true using the format of a 2-column proof.