This is the cinderella applet which shows that the angular bisector and the perpendicular bisector do not cross in the interior of the triangle, no matter where you put the vertices. you can play around by moving the vertices around by clicking and dragging.
Therefore, the problem with the false proof was that when we first drew the picture we assumed that the angular bisector and the perpendicular bisector would cross in the interior of the triangle. However it does not as can be seen from this applet.
Created with Cinderella
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