BID for the angle

Geometry Level 5

Given in the figure, a \(\Delta ABC\) with sides \(AB=3\sqrt{2}-\sqrt{6}\), \(AC=3\sqrt{2}+\sqrt{6}\) and \(BC=6\) units. \(AD\) is the angle bisector of \(\angle BAC\), with \(D\) on \(BC\). \(I\) is the in-center of \(\Delta ABC\).

\[\large{\angle BID=\cot^{-1} \left(\sqrt{P}+\sqrt{Q}-\sqrt{R}-S \right)}\]

where \(P,Q,R,S\) are integers with \(P,Q,R\) being square free. Find the value of \(P+Q+R+S\).

Clarification: \(\cot^{-1} (x)=\text{arccot} (x)\).

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