The Special Point in a Triangle 2

Geometry Level 5

Let OO be a point in acute-angle triangle ABCABC.

DD is the intersection of AOAO and BCBC. E,FE,F are defined similarly.

XX is the intersection of EFEF and ADAD. Y,ZY,Z are defined similarly.

Let PP be the intersection of XYXY and CFCF and QQ be the intersection of XZXZ and BEBE.

RR is the intersection of APAP with BCBC and SS is the intersection of AQAQ with BCBC.

When OO is the circumcentre, BRRC=439\frac{BR}{RC}=\frac{4 \sqrt{3}}{9}.

When OO is the orthocentre, BRRC=233\frac{BR}{RC}=\frac{2 \sqrt{3}}{3}.

Find 4×CB\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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