To determine in which situation the circumcenter always lies outside of a triangle, let's explore the geometric properties of the triangle and its circumcenter:
### Circumcenter and Its Properties:
1. Definition: The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. It is equidistant from all three vertices of the triangle.
### Location of the Circumcenter:
- Acute Triangle: In an acute triangle, all the angles are less than 90 degrees. The circumcenter lies inside the triangle.
- Right Triangle: In a right triangle, one of the angles is exactly 90 degrees. The circumcenter lies on the hypotenuse.
- Obtuse Triangle: In an obtuse triangle, one of the angles is greater than 90 degrees. The circumcenter lies outside the triangle.
- Isosceles Triangle: An isosceles triangle has two sides of equal length. Depending on the specific measures of the angles, the circumcenter can lie inside, on, or outside the triangle, but the criterion we're interested in is whether it is always outside.
### Conclusion:
- The circumcenter always lies outside the triangle if the triangle is obtuse.
Thus, the correct answer is:
- When the triangle is obtuse.
