Heron's Formula - Proof

Proof

A modern proof, which uses algebra and is quite unlike the one provided by Heron (in his book Metrica), follows. Let a, b, c be the sides of the triangle and A, B, C the angles opposite those sides. We have

by the law of cosines. From this proof get the algebraic statement:

The altitude of the triangle on base a has length b·sin(C), and it follows

\begin{align} T & = \frac{1}{2} (\mbox{base}) (\mbox{altitude}) \\ & = \frac{1}{2} ab\sin \widehat C \\ & = \frac{1}{4}\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2} \\ & = \frac{1}{4}\sqrt{(2a b -(a^2 +b^2 -c^2))(2a b +(a^2 +b^2 -c^2))} \\ & = \frac{1}{4}\sqrt{(c^2 -(a -b)^2)((a +b)^2 -c^2)} \\ & = \sqrt{\frac{(c -(a -b))(c +(a -b))((a +b) -c)((a +b) +c)}{16}} \\ & = \sqrt{\frac{(b + c - a)}{2}\frac{(a + c - b)}{2}\frac{(a + b - c)}{2}\frac{(a + b + c)}{2}} \\ & = \sqrt{\frac{(a + b + c)}{2}\frac{(b + c - a)}{2}\frac{(a + c - b)}{2}\frac{(a + b - c)}{2}} \\ & = \sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}. \end{align}

The difference of two squares factorization was used in two different steps.

Read more about this topic:  Heron's Formula

Famous quotes containing the word proof:

    “If some books are deemed most baneful and their sale forbid, how, then, with deadlier facts, not dreams of doting men? Those whom books will hurt will not be proof against events. Events, not books, should be forbid.”
    —Herman Melville (1819–1891)

    “He who has never failed somewhere, that man can not be great. Failure is the true test of greatness. And if it be said, that continual success is a proof that a man wisely knows his powers,—it is only to be added, that, in that case, he knows them to be small.”
    —Herman Melville (1819–1891)

    “Right and proof are two crutches for everything bent and crooked that limps along.”
    —Franz Grillparzer (1791–1872)