If D, E and F are the feet of the angle bisectors of ΔABC, then what do you notice about the incentre I of ΔABC, the orthocentre H and circumcentre O of ΔDEF? Drag any of the vertices A, B or C to dynamically move and change the figure to check your observation.
Though it might seem at first glance as if these 3 points are collinear, it should quickly become apparent by dragging to extreme cases that they are not collinear in general. Recall that for a result in mathematics to be true it has to be valid for all cases; so the conjecture is false.
However, what do you notice as you drag ABC? Can you modify your conjecture to a special type of triangle? Check your guess at Collinear Conjecture 2.
Michael de Villiers, 9 July 2011.