
Re: A Theorem concerning the Trisectors of a Triangle
Posted:
Sep 15, 1998 10:06 AM


On 12 Sep 1998, Den Roussel wrote (and I've edited for greater clarity):
> If the Angular trisectors of a triangle are produced to the Circumcircle, > then the Chords determined by the pairs of trisectors adjacent to the > edges 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 ?
As I've already remarked, this is indeed connected to Morley's theorem, in that Den Roussel's triangle is parallel to Morley's. However, I can produce no deduction of this result from Morley's theorem that's any simpler than an outright proof of it not using that theorem.
I conjectured that the parallelism might continue to hold if the trisectors of the typical angle (say B) were replaced by arbitrary pairs of isogonal lines (ie., B B1 and B B2 such that angles A B B1 and C B B2 are equal). The main purpose of this note is to announce that I've now disproved this conjecture (by replacing the Morley triangle by the Brocard "minor triangle").
However, I'm still very hopeful that there will exist SOME wonderful generalization, because this construction is near others that have interesting properties. For instance, it is known that if A(P), B(P), C(P) are the points where the Cevians of P hit the circumcircle again, then the triangle they form is similar to the pedal triangle of P (whose vertices A[P], B[P], C[P] are where the normals from P hit the sides).
John Conway

