Bri's Conjecture

The diagram below represents the argument that Bri made in class to demonstrate the truth of the Angle Bisector Theorem's claim. Her claim is that the relationship between the side of the parallelogram and its diagonal is the same as the other parallelogram and its diagonal, given the fact that the two parallelograms' corresponding angles are congruent. Play with the applet. Points A, B, and C are movable. Think about the theorem, which we now know is true through the Side Splitter Theorem. Do you think that her conjecture is true and provable, independent of the theorem, or do you think that it is dependent on it? In other words, is her conjecture provable without utilizing the Angle Bisector theorem?