Investigation: Does the 1st De Villiers point exist in hyperbolic geometry?

Consider the sketch below where G (not shown) is the incentre of the hyperbolic triangle ABC on the Poincare' disk, and D, E and F are the respective incentres of ABG, BCG and CAG. Drag any of the vertices to investigate whether AE, BF and CD are always concurrent. What do you notice? Can you prove (or refute) your observation?

Please enable Java for an interactive construction (with Cinderella). Below is a static sketch if the Java applet doesn't work.

Hyperbolic De Villiers1 Test

Created by Michael de Villiers, 25 July 2010 with Cinderella