Integral Median

$A B C$ is an integral sided triangle with sides $a \leq b \leq c$ .
$m_{C}$ is the median connecting $C$ and the midpoint of $A B$ .
$F (n)$ is the number of such triangles with $c \leq n$ for which $m_{C}$ has integral length as well.
$F (10) = 3$ and $F (50) = 165$ .

Find $F (100000)$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.