On 9/14/11, Eugene Salamin <gene_salamin@yahoo.com> wrote:
Fred Lunnon: ... let S be the fourth point within the original triangle T, the Cevian lines through S partitioning T into six subsidiary triangles. It looks to me as if not only
(1) When S is the incentre of T, then the incentres of the six sub-triangles lie on a conic;
but also
(2) When S is the centroid of T, then the centroids of the six sub-triangles lie on a conic. ----------
The proposition (2) is invariant under affine transformations. There exists an affine transformation that takes T into an equilateral triangle, in which case, the six centroids lie on a circle.
-- Gene
Neatly puncturing an earlier claim of mine that affine geometry was no use! But while this also gives a way to define corresponding Cevian points in distinct triangles, and preserves conics, of course it fails to preserve incentres and other special points. In the plane, the only transformations which preserve all lines are projective transformations; Moebius transformations preserve angle, and their intersection with the projectivities is just the Euclidean similarities. I don't actually know that other conformal plane transformations --- such as z -> exp(z) --- cannot ever be projectivities, but it looks plausible. However I've often wondered whether there exist more subtle plane transformations which preserve (say) some lines but not all, together with some angles but not all, which might attack the in-centre case in the same manner as Gene's one-liner for the centroid case? Fred Lunnon