Equivalence of Definitions of Heronian Triangle

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Theorem

The following definitions of the concept of Heronian Triangle are equivalent:

Definition 1

A Heronian triangle is a triangle whose side lengths and area are all rational numbers.

Definition 2

A Heronian triangle is a triangle whose side lengths and altitudes are all rational numbers.


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 if and only if $h_a$, $h_b$ and $h_c$ are rational.

Hence the result.

$\blacksquare$


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