Two One-Line Proofs of Heron’s Formula that says that the Area-Squared of a Triangle whose Side-Lengths are a,b,c is (a+b+c)(a+b-c)(a+c-b)(b+c-a)/16

Shalosh B. EKHAD and Doron ZEILBERGER^{1}

Maple Proof (by SBE): In a Maple session type:

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 0. seconds!): true.

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 · y/2 = y/2.

Human Proof (by DZ): The area-squared is obviously a symmetric and 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.

Comments: 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 lastbut-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!

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

References

[D] William Dunham, Heron, Newton, Euler, and Barney, MAA invited talk, JMM, Baltimore,Saturday January 18, 2014, 10:05 a.m.-10:55 a.m,

abstract: http://jointmathematicsmeetings.org/amsmtgs/2160 abstracts/1096-a0-12.pdf
[W] Wikipedia, Heron’s Formula, http://en.wikipedia.org/wiki/Heron

1 Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen

Rd., Piscataway, NJ 08854-8019, USA. zeilberg at math dot rutgers dot edu , http://www.math.rutgers.edu/~zeilberg/ . Jan. 20, 2014. Supported in part by the NSF.