Date: Sat, 27 Jul 1996 23:33:04 -0500 (CDT)
From: [Permission pending]
To: rusin@math.niu.edu
Subject: Re:  morley and all that

[deletia -- djr]

Proofs of Morley's theorem aren't difficult to locate, for
the few folks who still use mathematics libraries instead
of asking others for help.  Many "college geometry" texts
give proofs, including Coxeter.  I don't recall whether it
is proved by R. Johnson or N. A. Court, but the odds are
very good that a clean, clear argument is given by at least
one, if not by both.

I once heard a seminar talk given by J. W. Peters, perhaps
20 years ago, in which he explain the context of Morley's
theorem and showed how Morley came upon it.  My memory of
the details is completely gone, but I recall that there is
much more interesting geometry nearby.  There is a well
studied way of doing elementary analytic geometry using
complex notation and language (ultimately based on the fact
that  2x = z + z*  and  2y = z - z*  (writing  *  for the
complex conjugate).  In this notation there are ways of
handling lines, slopes, circles, etc., and in particular the
notion of slope has a meaningful complex extension.  Morley's
result pops out of a perfectly natural question in this context.

If you are interested, I'll try to ferret out a reference--
I seem to remember an expository article somewhere, perhaps
in the Monthly or Elemente der Mathematik, sometime in the last
30 years--but it may take me a while to locate it because I do
not recall either the title or the author or the journal for sure.

[deletia -- djr]

==============================================================================
[I did find this article in MathSciNet: -- djr]

   Mathematical Reviews on the Web
   3,251e 48.0X
   Peters, J. W.
   The theorem of Morley. 
   Nat. Mat. Mag. 16, (1941). 119--126.
   © Copyright American Mathematical Society 1942, 1998