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Geometry Level 5

Consider a \(\Delta ABC\) with \(\angle B=75^{\circ}\). Let \(H\) and \(O\) be its \(\text{orthocenter and circumcenter respectively}\). \(AD\bot BC\) with \(D\) on \(BC\). Also \(AH\times HD=6(\sqrt{3}-1)\) and circumradius of the \(\Delta ABC\) is \(2\sqrt{3}~units\).

\[\large{\angle HBO=\arcsin \left(\frac{P}{Q}(\sqrt{R}-\sqrt{S}) \right)}\]

where \(P,Q,R,S\) are integers and \(P,Q\) are co prime and \(R,S\) are square free.

Find \(P+Q+R+S\)

Original problem

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