Equivalence of Definitions of Heronian Triangle

Proof
Let $\triangle ABC$ be a triangle whose sides $a$, $b$ and $c$ are rational in length.

Let $\AA$ be the area of $\triangle ABC$.

Let $h_a$, $h_b$ and $h_c$ be the altitudes from $A$, $B$ and $C$ respectively.

From Area of Triangle in Terms of Side and Altitude, we have:


 * $\AA = \dfrac {c \cdot h_c} 2 = \dfrac {b \cdot h_b} 2 = \dfrac {a \cdot h_a} 2$

We are given that $a$, $b$ and $c$ are rational.

It follows from Rational Multiplication is Closed and Rational Division is Closed that $\AA$ is rational $h_a$, $h_b$ and $h_c$ are rational.

Hence the result.