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 \(\lfloor X \times 1000 \rfloor\)

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