Collinear Conjecture

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.

Please install Java (version 1.4 or later) to use JavaSketchpad applets.

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.

This page uses JavaSketchpad, a World-Wide-Web component of The Geometer's Sketchpad. Copyright © 1990-2011 by KCP Technologies, Inc. Licensed only for non-commercial use.

Michael de Villiers, 9 July 2011.