<< Twice again my messages have been bounced back, so math-fun don't know that the problem has been debunked by Stan Rabinowitz, using Geometer's Sketchpad. R. On Thu, 8 Sep 2011, Dan Asimov wrote:
Richard Guy, who is having trouble posting to math-fun, sent this message:
-------------------------------------------------------------------------- Apologies for butting in, but my messages to math-fun are bounced back by a spam filter, so I don't know how to send. What I wanted to say was:
The following problem was given to me by Julian Salazar, a high school student, who said he got it second-hand from members of the US IMO team. There are several subsidiary problems.
The medians of a triangle divide it into six smaller triangles. Show that the six incentres lie on an ellipse.
1. Who originated the problem?
2. Did he/she have a proof?
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.
4. In the original problem, is the centre of the ellipse among Clark Kimberling's three-thousand-odd triangle centres?
5. Same question as 4., but taking the defining point to be other of the more familiar triangle centres, besides the centroid.
6. If we apply Conway's extraversion to the problem, are there any selections of six excentres which lie on a conic?
I must be getting old, because any attempt at finding a proof bogs down in a morass between Euclidean and projective geometry. Or, just possibly, it isn't true?!?
Happy new academic year to all. R.
Sometimes the brain has a mind of its own.