Heron's Formula/Proof 1

Theorem
Given a triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite points $A$, $B$, and $C$, respectively.

Let $s$ be the semiperimeter, so $s = \dfrac{a + b + c} 2$.

Then the area $A$ of the triangle is given by the formula $A = \sqrt{s \left({s - a}\right) \left({s - b}\right) \left({s - c}\right)}$.

Proof
Construct the altitude from $A$. Let the length of the altitude be $h$ and the foot of the altitude be $D$.

Let the distance from $D$ to $B$ be $z$.


 * Heron's Formula.png

Then $h^2 + \left({a - z}\right)^2 = b^2$ and $h^2 + z^2 = c^2$ from Pythagoras's Theorem.

By subtracting these two eqns, we get $2az - a^2 = c^2 - b^2$, which simplifies to $z = \dfrac{a^2 + c^2 - b^2}{2a}$.

Plugging back in and simplifying yields $h = \sqrt{c^2 - \left({\dfrac{a^2 + c^2 - b^2}{2a}}\right)^2}$, and so: