# 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.

The Special Point In A Triangle 1

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