Equivalence of Definitions of Heronian Triangle
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.
$\blacksquare$
Sources
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Bachet