Deceptively Complex

Geometry Level 5

In an Argand diagram, $$O$$ is the origin and $$P$$ is the point $$2+0i$$. The points $$Q, R$$ and $$S$$ are such that the length of $$OP, PQ, QR$$ and $$RS$$ are equal, and that $$\angle OPQ = \angle PQR = \angle QRS = \frac{5\pi}{6}$$. Note that the points $$O,P,Q,R,S$$ are $$5$$ vertices of a regular $$12$$-sided polygon lying on the upper half of the Argand diagram.

The point $$C$$ is the circumcenter of triangle $$OPQ$$. If the polygon is rotated anticlockwise about $$O$$ until $$C$$ first lies on the real axis, the new position of $$S$$ can be written as

$$( \large -\frac{A}{B}(\sqrt{C}+\sqrt{D}), \frac{E\sqrt{F}+\sqrt{G}}{H}i)$$,

where $$A,B,C,D,E,F,G,H$$ are not necessarily distinct positive integers with $$C,D,F,G$$ squarefree.

Find $$A+B+C+D+E+F+G+H$$.

This problem comes from STEP (Sixth Term Examinations Paper) year 2001.

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