Question

any obtuse triangle, which point of concurrency lies outside the triangle? select all that apply
incenter
a circumcenter
c orthocenter
d centroid

Explanation:

• Incenter: The incenter is the intersection of angle bisectors. In any triangle (acute, right, obtuse), it lies inside the triangle as it is the center of the incircle (tangent to all sides), so it can't be outside.
• Circumcenter: The circumcenter is the intersection of perpendicular bisectors. In an obtuse triangle, the circumcenter lies outside the triangle. For acute triangles, it's inside; for right triangles, it's on the hypotenuse.
• Orthocenter: The orthocenter is the intersection of altitudes. In an obtuse triangle, two of the altitudes are outside the triangle (from the acute vertices to the opposite extended sides), so their intersection (orthocenter) lies outside the obtuse triangle. For acute triangles, it's inside; for right triangles, it's at the right - angled vertex.
• Centroid: The centroid is the intersection of medians. In any triangle, the medians are always inside the triangle, so the centroid (which divides each median in a 2:1 ratio) is always inside the triangle.

Answer:

B. circumcenter, C. orthocenter