Heron's Formula/Proof 1

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 $z$.


 * Heron1.png

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

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

By subtracting $\paren 1$ from $\paren 2$:
 * $2 a z - a^2 = c^2 - b^2$

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

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

and so: