A median of a triangle through mid-point of $(-c,0),(c,0) $ is such so that ratio of cosines of angles between sides/median
$$ \cos \phi/\cos \psi =e $$ is a constant. Is the curve known?
$$ \frac {((x + c) x + y^2)}{((x - c) x + y^2)} \, \sqrt{ \frac{((x - c) ^2 + y^2}{((x + c) ^2 + y^2}} = e $$
Seems to be a fourth order curve: