# The Orthocenter Of Triangle Joining The Circumcenters

Geometry Level 4

Let $$O$$ be the circumcenter of an acute $$\triangle ABC.$$ Let $$O_A, O_B, O_C$$ be the circumcenters of $$\triangle BCO, \triangle CAO, \triangle ABO$$ respectively, and let $$S$$ be the circumcenter of $$\triangle O_AO_BO_C.$$ Let $$H$$ be the orthocenter of $$\triangle ABC.$$ Find $$\angle OSH \pmod{ \pi}$$ in degrees.

Details and assumptions

• The notation $$\angle OSH \pmod{\pi}$$ means you have to enter the remainder when $$\angle OSH$$ is divided by $$180$$ in degrees. For example, if you think $$\angle OSH = 275^{\circ},$$ you should enter $$95.$$ In particular, if you think $$\angle OSH = 180^{\circ},$$ (that is, $$O,S,H$$ are collinear), enter $$0.$$

• The image shown is not accurate.

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