Integer Heronian Triangle can be Scaled so Area equals Perimeter

Theorem

Let $T_1$ be an integer Heronian triangle whose sides are $a$, $b$ and $c$.

Then there exists a rational number $k$ such that the Heronian triangle $T_2$ whose sides are $k a$, $k b$ and $k c$ such that the perimeter of $T$ is equal to the area of $T$.

For a given triangle $T$:

let $\map \AA T$ denote the area of $T$

let $\map P T$ denote the perimeter of $T$.

We are given that $T_1$ is an integer Heronian triangle whose sides are $a$, $b$ and $c$.

Let $\map P {T_1} = k \map \AA {T_1}$.

Let $T_2$ have sides $k a$, $k b$ and $k c$.

Then we have that:

$\ds \map P {T_2}$

$=$

$\ds k \map P {T_1}$

$\ds \map A {T_2}$

$=$

$\ds k^2 \map A {T_1}$

$\ds $

$=$

$\ds k \map P {T_1}$

$\ds $

$=$

$\ds \map P {T_2}$

$\blacksquare$