> Here is another Morley triangle oddity. Take the six triangles around the > Morley triangle (formed by not erasing the segments from the vertices of > the original triangle to the morely triangle) and call the respective > incenters of these six triangles A, B, C, D, E, F (moving say clockwise > around). Then the three segments AD, BE and FC are concurrent. (I don't > know how to prove this). The same is true for circumcenters and centroids > (but not orthocenters). The proof for centroids is easy--infact it has > nothing to do with the Morley triangle (any other triangle will do). As for > the others, I have not found any proofs.
I think I can see my way to proving at least some of these results, but only using barycentric coordinates. Anyway, I'll have a try. John Conway