Do you like ellipse?

Project the plane such that the ellipse becomes the circle, in this projection ratio of areas are invariant!
$\\ \implies k= (\frac{\pi r^2}{\Delta} ) =(\frac{\pi \Delta^2}{s^2 \Delta} ) =(\frac{\pi \Delta}{s^2})$
w.l.o.g assume s=1 $\implies$ a+b+c=2
$\implies \pi \Delta= \pi \sqrt((1-a)(1-b)(1-c)) \leq \pi (\frac{(1-a)+(1-b)+(1-c)}{3})^\frac{3}{2} = \frac{\pi}{3\sqrt3} \space using \space AM-GM$ $\\ \implies MAX \space k = \frac{\pi}{3\sqrt3} \approx 0.60$

link text
This is known as Steiner_inellipse and the above link gives us that the ratio is $\dfrac \pi {3\sqrt3}.$
This link is from my solution to almost the same problem "Medial Ellipse" by Sharky Kesa, just two days back.

The ellipse is Steiner inellipse . The ratio is $\dfrac{\pi}{3\sqrt{3}}\approx \boxed{0.60}$