# 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 \(0.\)

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