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Turkish NMO First Round - 1999 P-29 (Geometry)

Source:

July 3, 2012
geometryratiotrapezoidpower of a point

Problem Statement

The length of the altitude of equilateral triangle ABC ABC is 33. A circle with radius 22, which is tangent to [BC] \left[BC\right] at its midpoint, meets other two sides. If the circle meets AB AB and AC AC at D D and E E, at the outer of △ABC\triangle ABC , find the ratio Area (ABC)Area (ADE) \frac {Area\, \left(ABC\right)}{Area\, \left(ADE\right)}.
(A)\ 2\left(5 \plus{} \sqrt {3} \right) \qquad(B)\ 7\sqrt {2} \qquad(C)\ 5\sqrt {3} \\ \qquad(D)\ 2\left(3 \plus{} \sqrt {5} \right) \qquad(E)\ 2\left(\sqrt {3} \plus{} \sqrt {5} \right)
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