Let the sides of triangle $T$ have lengths $a$, $b$ and $c$ and the
corresponding altitudes have lengths $H_a$, $H_b$ and $H_c$.
By the triangle inequality, we have $a+b > c$, $b+c > a$ and
$c+a > b$ and so $a+b+c > 2c$, $a+b+c > 2a$ and $a+b+c
> 2b$.
Also, since $T$ has area $1$, we have $\frac{1}{2}aH_a=1$,
$\frac{1}{2}bH_b=1$ and $\frac{1}{2}cH_c=1$ and so $aH_a=2$,
$bH_b=2$ and $cH_c=2$.
$$M=(a+b+c)(H_a+H_b+H_c)=(a+b+c)H_a+(a+b+c)H_b+(a+b+c)H_c$$ $$ >
2aH_a+2bH_b+2cH_c=4+4+4=12$$ so $M> 12$ and hence statement
E is false.