BdMO(Regional) 2012 Problem

Geometry Level 3

In a circle, \(AB=4\) is the diameter and \(O\) is the centre. \(C\) and \(D\) are chosen such that \(C,O,D\) are co-linear, \(\angle COB= \angle AOD=30^\circ\), \(OC=OD\) and \(\angle BCO = 90^\circ\). The perpendicular on \(AB\) through \(O\) meets \(AC\) at \(E\) and \(BD\) at \(F\).

If \(EF=\frac{a\sqrt{b}}{c}\) where \(a,b,c\) are integers and \(b,c\) are primes, find the value of \(a+b+c\).

Note: \(C\) and \(D\) are not necessarily on the circumference of the circle.

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