## Collinear Conjecture

If *D*, *E* and *F* are the feet of the angle bisectors of Δ*ABC*, then what do you notice about the incentre *I* of Δ*ABC*, the orthocentre *H* and circumcentre *O* of Δ*DEF*? Drag any of the vertices *A*, *B* or *C* to dynamically move and change the figure to check your observation.

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Collinear Conjecture

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.

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Michael de Villiers, 9 July 2011.