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\bf
\centerline
{
Two One-Line Proofs of Heron's Formula  that says that the Area-Squared of a Triangle 
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\centerline
{
whose Side-Lengths are a,b,c is (a+b+c)(a+b-c)(a+c-b)(b+c-a)/16
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\rm
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\centerline{ {\it
Shalosh B. EKHAD and
Doron 
ZEILBERGER}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019, USA.
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{\eighttt zeilberg  at math dot rutgers dot edu} ,
\hfill \break
{\eighttt http://www.math.rutgers.edu/\~{}zeilberg/} .
Jan. 20, 2014.
Supported in part by the NSF.
}
}

{\bf Maple Proof} (by SBE): In a Maple session type:

{\eighttt evalb(expand(subs({c=1,b=sqrt(x\^{}2+y\^{}2),a=sqrt((x-1)\^{}2+y\^{}2)},(a+b+c)*(a+b-c)*(a+c-b)*(b+c-a)/16-(y/2)\^{}2))=0);} 

and get (after  {\tt 0.} seconds!): {\tt true}.

{\it Clarification}: Without loss of generality the vertices of the triangle are at $A(0,0)$, $B(1,0)$, and $C(x,y)$.
So $c=1$, $b=\sqrt{x^2+y^2}$,  $a=\sqrt{(x-1)^2+y^2}$, and the area is $1 \cdot y/2=y/2$.

{\bf Human Proof} (by DZ): The area-squared is obviously a {\it symmetric} and {\it homogeneous} polynomial 
of degree $4$ in $a,b,c$, divisible by $(a+b-c)(a+c-b)(b+c-a)$, since degenarate triangles have zero area.
Hence the area-squared divided by  $(a+b-c)(a+c-b)(b+c-a)$ is a symmetric and homogeneous polynomial of degree $1$ in
$a,b,c$, and so is $(a+b+c)$ times some constant that must be $\frac{1}{16}$ by considering, say, the $90^o,45^o,45^o$ triangle.

{\bf Comments}: {\bf 1.} 
These two proofs were inspired by the excellent MAA invited talk[D] by William Dunham,  where  I was astonished on how
such great people as Hero, Newton, and Euler, and last-but-not least, the great inventor Bernard (``Barney'') Oliver
gave complicated proofs to this utter triviality.
I am sure that all the human-generated mathematics published today, by great people such as Andrew Wiles and Yitang Zhang,
would be considered utterly trivial in a hundred years. Our only hope at non-trivial mathematics is
to use computers!

{\bf 2.} The above two proofs are much shorter (and nicer!) than all the proofs in [W].


{\bf References}

[D] William Dunham, {\it Heron, Newton, Euler, and Barney}, MAA invited talk, JMM, Baltimore,Saturday January 18, 2014, 10:05 a.m.-10:55 a.m,
\hfill\break
abstract: {\tt http://jointmathematicsmeetings.org/amsmtgs/2160\_abstracts/1096-a0-12.pdf}

[W] Wikipedia, {\it Heron's Formula}, {\tt http://en.wikipedia.org/wiki/Heron%27s\_formula} .

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