No 1847 Sangaku Problem

In the large circle, five circles - four large green and one small orange - are mutually tangent to each other. As shown above, two green circles are tangent to the base of the inscribed isosceles triangle, containing three circles. Two cyan ellipses, both of the maximum possible eccentricity, are each tangent to the leg of the triangle as well as the circumference of the large circle - each side at one point. In addition, their major axes are parallel to their respective leg of the triangle.

If the following ratio can be expressed as

$\dfrac{\text{major axis of the ellipse}}{\text{diameter of the orange circle}} = \dfrac{A}{B}\left(C + D\sqrt{E}\right)$

where $A,B,C,D,E$ are positive integers, $\gcd(A,B) = \gcd(C,D) = 1$ and $E$ is square-free, input $A + B + C + D + E$ as your answer.