Tangent ellipses

Geometry Level 5

There is a circle of a radius $$\sqrt{\frac{8}{3}}$$. There are 3 ellipses inside the circle which are tangent to the circle at points $$A, B, C$$ and to each other at points $$D, E, F$$. Their major axes are parallel to lines tangent to the circle at corresponding tangency points. The lengths of major axes of the ellipses are:
$$2 \sqrt{\frac{3}{2}}$$, $$2 \sqrt{\frac{8}{7}}$$ and $$2$$. $X = \frac{\triangle ABC}{ \triangle DEF}$

Find $$\lfloor X \times 1000 \rfloor$$

×