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.
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.
Related Links
Isosceles Triangle Collinear Conjecture
Triangle Incentre-Circumcentre Collinearity
Some Trapezoid (Trapezium) Explorations (see Investigations 3 & 4)
An interesting collinearity
Euler line proof
Further Euler line generalization
Euler-Nagel line analogy
Nine Point Conic and Generalization of Euler Line
Spieker Conic and generalization of Nagel line
More Properties of a Bisect-diagonal Quadrilateral
Quadrilateral Similar Triangles Collinearity
Investigating incentres of some iterated triangles
Investigating circumcentres of iterated median triangles
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Michael de Villiers, created 9 July 2011 with JavaSketchpad; updated to WebSketchpad, 11 July 2025.