Eccoci.

$ABC$ is an integral sided triangle with sides $a \le b \le c$.
$m_C$ is the median connecting $C$ and the midpoint of $AB$.
$F(n)$ is the number of such triangles with $c \le n$ for which $m_C$ has integral length as well.
$F(10)=3$ and $F(50)=165$.

Find $F(100000)$.