Den Roussel wrote: >While investigating the Morley triangle, I came upon this theorem which >some might find interesting. > > If the Angular trisectors of a triangle >are produced to the Circumcircle , then the Chords of adjacent >trisectors form an Equilateral triangle. > >It has been suggested that this result might follow from Morleys >theorem. So far, I have been unable to make this connection. Any >thoughts ? > >Den Roussel
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.