Another Special Theorem: Part 1 (V1)

Topic:
Angles
In the applet below, note that point E is equidistant from the SIDES of ANGLE BAC.     Directions: 1) Move the purple slider to adjust point 's distance from the sides of ANGLE BAC.      As you do, you'll notice that all possible locations of point E will be traced out.   2) What does the locus (set) of points in the plane equidistant from the sides of an angle     look like?   Be specific!   3) Now move points A and B around to change the initial measure of the displayed angle.     After doing so, hit the "clear trace" button to clear the previous traces of E.   4) Repeat step (1).  Does your response for (2) above still seem valid?   3) Use the tools of GeoGebra to show that your response in (2) above is true.
Use your observations from interacting with the applet above to complete the following statement:   If a point is ____________________ from the ____________ of an ______________, then that   __________________ lies on the ___________________ of that ________________.   Now prove this theorem true using a 2-column format.