# The Special Point in a Triangle 2

**Geometry**Level 5

Let \(O\) be a point in acute-angle triangle \(ABC\).

\(D\) is the intersection of \(AO\) and \(BC\). \(E,F\) are defined similarly.

\(X\) is the intersection of \(EF\) and \(AD\). \(Y,Z\) are defined similarly.

Let \(P\) be the intersection of \(XY\) and \(CF\) and \(Q\) be the intersection of \(XZ\) and \(BE\).

\(R\) is the intersection of \(AP\) with \(BC\) and \(S\) is the intersection of \(AQ\) with \(BC\).

When \(O\) is the **circumcentre**, \(\frac{BR}{RC}=\frac{4 \sqrt{3}}{9}\).

When \(O\) is the **orthocentre**, \(\frac{BR}{RC}=\frac{2 \sqrt{3}}{3}\).

Find \(\frac{4 \times \angle C}{\angle B}\) where we measure the angles in degrees.

**Note**:
Where intersections between sides are described it is assumed the sides are extended if necessary.

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