9 Sep
2011
9 Sep
'11
12:10 a.m.
3. Experiment suggests that if the three ``cevians'' are drawn through any point, not only the centroid, then the result is still true, except that if the point is outside the triangle, then the conic is a hyperbola, and if the point is on an edge of the triangle, the conic is a pair of straight lines.
Draw lines through the incentres of opposite triangles. They must concur at this point (the centroid in the original problem), as this is the common centre of homothety for the incircles. The result that they are conconic follows from Brianchon's Theorem. To prove 'ellipse' rather than 'conic', one need only show that the conic is bounded. I guess that it is contained within the original triangle. Sincerely, Adam P. Goucher