A triangle with one angle bigger than 90 degrees will have its circumscribing circle's centre outside that triangle.
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When the triangle is obtuse.
It doesn't ever fall outside of a triangle.
A point outside the triangle may.
The orthocenter may fall outside of a triangle. The orthocenter usually lies within the inside the triangle. However this is only the case if the triangle is acute.
In an obtuse angled triangle, two of them will.