I've looked at a number of proofs of Heron's formula and other related questions on this forum, but I was just wondering if the following half-baked idea could relate to yet another proof. Rearranging the formula:
$$\begin{align} K &= \sqrt{s(s-a)(s-b)(s-c)}\\[0.5em] &= s^2\frac{\sqrt{s(s-a)}}s\frac{\sqrt{s(s-b)}}s\frac{\sqrt{s(s-c)}}s \\[0.5em] &= s^2\frac{\sqrt{(b+c)^2-a^2}}{a+b+c}\frac{\sqrt{(a+c)^2-b^2}}{a+b+c}\frac{\sqrt{(a+b)^2-c^2}}{a+b+c} \end{align}$$
Now the first ratio, for example, is the ratio of final to initial height if you take a line segment of length $(a+b+c)$ and collapse it into a right triangle with base $a$ and hypotenuse $b+c$. So I'm thinking, maybe if you start with a triangle of base $2s$ and height $s$ or vice versa (hence area $s^2$), and perform three such sequential "collapse" operations, you could get the triangle $ABC$, while reducing the area by the corresponding ratio each time? Or something roughly in that vein...
I thought that could be nice because it would be a pretty short proof that more closely hews to the symmetry between $a$, $b$, and $c$.
