Heron's Formula/Proof 1

Theorem
Let $\triangle ABC$ be a triangle with sides $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.

Then the area $Q$ of $\triangle ABC$ is given by:
 * $Q = \sqrt{s \left({s - a}\right) \left({s - b}\right) \left({s - c}\right)}$

where $s = \dfrac{a + b + c} 2$ is the semiperimeter of $\triangle ABC$.

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

From Pythagoras's Theorem:
 * $(1): \quad h^2 + \left({a - z}\right)^2 = b^2$

and:
 * $(2): \quad h^2 + z^2 = c^2$

By subtracting $(1)$ from $(2)$:
 * $2az - a^2 = c^2 - b^2$

which can be expressed in terms of $z$ as:
 * $z = \dfrac{a^2 + c^2 - b^2}{2a}$

Substituting for $z$ in $(2)$ and simplifying yields:
 * $h = \sqrt{c^2 - \left({\dfrac{a^2 + c^2 - b^2}{2a}}\right)^2}$

and so: