Butterfly Theorem Action!

Topic:
Circle
Note: Creation of this applet was inspired by a tweet from Alexander Bogomolny (at Cut-the-Knot.) The applet below dynamically illustrates a theorem known as the Butterfly Theorem. Interact with this applet for a few minutes. As you do, feel free to change the location(s) of any 1 (or more) of the BIG POINT(S) that are already there (or will soon appear)! How can you formally prove what this applet informally illustrates?