Proof of Heron's Formula for the area of a triangle 
Let $a,b,c$ be the lengths of the sides of a triangle. The area is given by Heron's formula:
$$A = \sqrt{p(p-a)(p-b)(p-c)},$$
where $p$ is half the perimeter, or $p=\frac{a+b+c}{2}$.

Could you please provide the proof of this formula?
Thank you in advance.

It is actually quite simple. Especially if you allow using trigonometry, which, judging by the tags, you do. If $\alpha$ is the angle between sides $a$ and $b$, then it is known that
$$
\begin{align}
A &=  \frac{ab\sin \alpha}{2},\\
A^2 &= \frac{a^2b^2\sin^2 \alpha}{4}.
\end{align}
$$
Now, $\sin^2 \alpha = 1 - \cos^2 \alpha$, and you can find $\cos \alpha$ from the law of cosines:
$$
c^2 = a^2 + b^2 - 2ab \cos \alpha.
$$
You just find $\cos \alpha$ from this equality, plug it into the formula for $A$ above, and Heron's formula pops up as a result.
