Question

which is the only center point that lies on the edge of a triangle?
○ the incenter of a right triangle
○ the incenter of an obtuse triangle
○ the circumcenter of a right triangle
○ the circumcenter of an obtuse triangle

Explanation:

1. Recall the definitions of incenter and circumcenter:

• The incenter is the intersection of angle bisectors, always inside the triangle.
• The circumcenter is the intersection of perpendicular bisectors. For a right triangle, the circumcenter is at the midpoint of the hypotenuse (on the edge). For an obtuse triangle, the circumcenter is outside the triangle.

1. Analyze each option:

• Incenter of right/obtuse triangle: Inside the triangle, so not on the edge.
• Circumcenter of right triangle: At midpoint of hypotenuse (on the edge).
• Circumcenter of obtuse triangle: Outside the triangle.

Answer:

C. the circumcenter of a right triangle