To determine the situation in which the circumcenter of a triangle always lies outside the triangle, let's review the properties of different types of triangles and the location of their circumcenters.
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect. This point is equidistant from the three vertices of the triangle and serves as the center of the triangle's circumcircle (the circle that passes through all three vertices).
### Types of Triangles and Circumcenter Positions:
1. Acute Triangle:
- All angles are less than 90 degrees.
- The circumcenter lies inside the triangle.
2. Right Triangle:
- One angle is exactly 90 degrees.
- The circumcenter lies at the midpoint of the hypotenuse.
3. Obtuse Triangle:
- One angle is greater than 90 degrees.
- The circumcenter lies outside the triangle.
4. Isosceles Triangle:
- Two sides are equal.
- The circumcenter's location depends on whether the isosceles triangle is acute, right, or obtuse.
- If the isosceles triangle is acute, the circumcenter will be inside.
- If the isosceles triangle is right, the circumcenter will be at the midpoint of the hypotenuse.
- If the isosceles triangle is obtuse, the circumcenter will be outside.
Since the question asks about the situation where the circumcenter always lies outside the triangle, the correct choice is when the triangle is obtuse. In obtuse triangles, one of the angles always exceeds 90 degrees, positioning the circumcenter outside the triangle.
Therefore, the circumcenter of a triangle always lies outside the triangle when the triangle is obtuse.
