Heron's Formula and Brahmagupta's Generalization

Let a,b,c be the sides of a triangle, and let A be the area of the
triangle. Heron's formula states that A^2 = s(s-a)(s-b)(s-c),
where s = (a+b+c)/2. The actual origin of this formula is somewhat
obscure historically, and it may well have been known for centuries
prior to Heron. For example, some people think it was known to
Archimedes. However, the first definite reference we have to this
formula is Heron's. His proof of this result is extremely circuitious,
and it seems clear that it must have been found by an entirely different
thought process, and then "dressed up" in the usual synthetic form
that the classical Greeks preferred for their presentations.

Here's a much more straightforward derivation. Consider the general
triangle with edge lengths a,b,c shown below



We have a = u+v, b^2 = h^2+u^2, c^2 = h^2+v^2. Subtracting the
second from the third gives u^2-v^2 = b^2-c^2. Dividing both sides
by a = u+v, we have u-v = (b^2-c^2)/a. Adding u+v = a to both
sides and solving for u gives

a^2 + b^2 - c^2
u = -----------------
2a

Taking h = sqrt(b^2-u^2) we have

__________________________________
1 | / \ 2 / a^2 + b^2 - c^2 \ 2
A = ah/2 = --- | ( ab ) - ( ----------------- ) (1)
2 \| \ / \ 2 /

which is equivalent to Heron's formula. Factoring out 1/4, this gives
three different ways of expressing (2ab)^2 - (a^2+b^2-c^2)^2 as a
difference of two squares. Equivalently, it gives three different
factorizations of 16A^2, each of the form

16A^2 = [(a+b)^2 - c^2] [c^2 - (a-b)^2] (2)

Factoring each of these terms gives the explicitly symmetrical form

16A^2 = (a+b+c)(a+b-c)(c-a+b)(c+a-b) (3)

so if we define s=(a+b+c)/2 we can write equation (1) as
________________
A = /s(s-a)(s-b)(s-c) (4)

which is the area formula as given by Heron.

For an alternative derivation see Heron's Formula For Tetrahedrons.

Incidentally, if we factor ab out of the radical in equation (1) we
get
____________________________
ab | / a^2 + b^2 - c^2 \ 2
A = ah/2 = --- | 1 - ( ----------------- ) (5)
2 \| \ 2ab /

Notice that if we take either the edge "a" or "b" as the base of the
triangle, then the height is sin(q) where q is the angle between "a"
and "b", and of course the area of half the altitude times the base
(which is easily seen by considering the parallelogram), so we have
A = (ab/2) sin(q), which implies that the radical in (5) equals the
sine of the angle between "a" and "b". Furthermore, it implies that
the cosine is given by the well-known formula

/ a^2 + b^2 - c^2 \
cos(q) = ( ----------------- )
\ 2ab /

[By the way, permutations of {a,b,c} = {3,5,7} in equation (2) give
the three factorizations 675 = (15)(45) = (9)(75) = (5)(135), which
leaves out (1)(675), (3)(225), and (25)(27). Is there an expression
in a,b,c that gives these three factorizations under permutation?]

One of the most beautiful things about Heron's formula is the
generalization discovered by the Hindu mathematician Brahmagupta
around 620 AD. He noted that we have a symmetrical product of four
factors inside the square root of equation (4), consisting of twice
the quantities

a+b+c a+b-c a-b+c -a+b+c

In a sense we can "full out" the symmetry, making each of the four
factors symmetrical with the others, by imagining a fourth "side"
of length d=0 being subtracted from the first factor and added to
the remaining three, so we have

a+b+c-d a+b-c+d a-b+c+d -a+b+c+d

Obviously with d=0 this is identical to the previous set of factors,
but it has greater formal symmetry, so it seems as if the quantity

1 _____________________________________
--- /(a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)
4

MUST represent... something meaningful. Indeed it does. This is the
area of a quadrilateral with sides a,b,c,d inscribed in a circle, i.e.,
a cyclic quadrilateral. Naturally every triangle is cyclic, meaning
that it can be inscribed in a circle, and a triangle can be regarded
as a quadrilateral with one of its four edge lengths set equal to zero.
Brahmagupta didn't actually give a formal proof of this result, and
in fact the surviving copies of his statement of this proposition
don't mention the fact that it applies only to cyclic quadrilaterals.
It's tempting to think that Brahmagupta might have just imagined the
equation based on its formal symmetry.

Incidentally, the formula for the area of an arbitrary quadrilateral
is

1 ________________________________________________________
--- /(a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d) - 16 abcd cos(q)^2
4

where q is half the sum of two opposite angles. For a cyclic
quadrilateral the each pair of opposite angles sums to pi, so it
reduces to Brahmagupta's formula.