Proof Using The Pythagorean Theorem
Heron's original proof made use of cyclic quadrilaterals, while other arguments appeal to trigonometry as above, or to the incenter and one excircle of the triangle . The following argument reduces Heron's formula directly to the Pythagorean theorem using only elementary means.
We wish to prove 4T2 = 4s(s - a)(s - b)(s - c). The left-hand side equals
while the right-hand side equals
via the identity (p + q)2 - (p - q)2 = 4pq. It therefore suffices to show
and
Substituting 2s = (a + b + c) into the former,
as desired. Similarly, the latter expression becomes
Using the Pythagorean theorem twice, b2 = d2 + h2 and a2 = (c - d)2 + h2, which allows us to simplify the expression to
The proof follows.
Read more about this topic: Heron's Formula
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