Thebault Action!!!

The following applet illustrates Thebault's Theorem without words, segment lengths, or angle measures. Interact with the following applet for a few minutes. Be sure to change the locations of the quadrilateral's vertices each time before sliding the slider. Without researching Thebault's Theorem on another tab in your internet browser, try to express this theorem in your own words as a conditional ("if-then") statement. How could you formally prove what this applet informally illustrates?
Creation of this applet was inspired by a tweet from Cliff Pickover.