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\def\halmos{\hbox{\vrule height0.15cm width0.01cm\vbox{\hrule height
 0.01cm width0.2cm \vskip0.15cm \hrule height 0.01cm width0.2cm}\vrule
 height0.15cm width 0.01cm}}
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\bf
\centerline
{A Proof of Morley's Theorem (From The (Devil's) Book)}
\rm
\bigskip
\centerline{ {\it Shalosh B. Ekhad}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Temple University,
Philadelphia, PA 19122, USA. 
%\break
e-mail:{\eighttt ekhad@math.temple.edu .}
www:{\eighttt http://www.math.temple.edu/$\tilde{\,\,}$ekhad   .}
} 
}
 
According to Erd\"os P\'al, God(the SF) has a notebook that contains
short and insightful proofs of all theorems. Don Newman([N]) has just
found such a proof of Morley's([M]) theorem:
 
{\bf Theorem ([M]):} If you trisect\footnote{$^2$}
{\eightrm In a lovely talk, delivered by Newman on 3/29/95, in the
Temple number theory seminar, he said that he was attracted
to this statement (for the last 50 years) because of the kick
that he derived from the apparent(?) illicitude of the act
of trisection.}
each angle of an arbitrary triangle,
and continue the trisection rays indefinitely, then the triangle
formed from the intersections of these rays is equilateral.
 
According to Doron Zeilberger, the devil(SC(?)), also has
a notebook, that contains purely routine, albeit often long,
proofs of all theorems. In this note we give such a proof.
 
{\bf  Proof:} {\tt maple <CR> read mephisto: morley(ta,tb);<CR>
quit} \quad \halmos
 
For the sake of completeness here is file {\tt mephisto}.
 
{\eightrm
f:=proc(a,b):(a+b)/(1-a*b):end: \quad \quad
f2:=proc(a);normal(f(a,a)):end: \quad \quad
f3:=proc(a);normal(f(a,f2(a))):end:
 
morley:=proc(ta,tb):
eq1:=y=x*ta: eq2:=y=(x-1)*(-tb): Eq1:=y=x*f3(ta): Eq2:=y=(x-1)*(-f3(tb)):
sol:=solve({Eq1,Eq2},{x,y}): Dx:=subs(sol,x):Dy:=subs(sol,y):
sol:=solve({eq1,eq2},{x,y}): Ax:=subs(sol,x):Ay:=subs(sol,y):
eq3:=y=x*f2(ta): eq4:=(y-Dy)=(x-Dx)*f(f(f2(ta),-tb),sqrt(3)):
sol:=solve({eq3,eq4},{x,y}): Bx:=subs(sol,x):By:=subs(sol,y):
eq6:=y=-(x-1)*f2(tb): eq5:=(y-Dy)=(x-Dx)*(-1)*f(f(f2(tb),-ta),sqrt(3)):
sol:=solve({eq5,eq6},{x,y}): Cx:=subs(sol,x):Cy:=subs(sol,y):
ABsq:=(Ax-Bx)**2+(Ay-By)**2: ACsq:=(Ax-Cx)**2+(Ay-Cy)**2: BCsq:=(Bx-Cx)**2+
(By-Cy)**2:
normal(ABsq-BCsq),normal(ACsq-BCsq);
end:
 
}
{\bf References}
 
[M] F. Morley, {\it ``Geometry''}, John Hopkins Univ. Press, 1904.
 
 
[N] Donald J. Newman, {\it A new proof of Morley's theorem}, Math. Intell.,
to appear.
\bye
