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 lies at the midpoint of the hypotenuse (on the edge). For an obtuse triangle, the circumcenter lies outside the triangle.

1. Analyze each option:

• Incenter of right/obtuse triangle: Incenter is always inside the triangle, so these are incorrect.
• Circumcenter of right triangle: Lies at midpoint of hypotenuse (on the edge), correct.
• Circumcenter of obtuse triangle: Lies outside the triangle, incorrect.

Answer:

C. the circumcenter of a right triangle