Question

for any obtuse triangle, which point of concurrency lies outside the triangle?
select all that apply.

☐ centroid

☐ incenter

☐ orthocenter

☐ circumcenter

Explanation:

1. Centroid: The centroid is the intersection of medians. In any triangle (including obtuse), medians are inside, so centroid is inside.
2. Incenter: The incenter is the intersection of angle bisectors, always inside the triangle (as it's the center of the incircle, tangent to all sides).
3. Orthocenter: In an obtuse triangle, the altitudes from the acute vertices will intersect the extended opposite sides. The orthocenter (intersection of altitudes) lies outside the triangle (opposite the obtuse angle).
4. Circumcenter: The circumcenter is the intersection of perpendicular bisectors. In an obtuse triangle, the perpendicular bisectors of the sides opposite the acute angles will intersect outside the triangle (as the circumcircle has its center outside for obtuse triangles).

Answer:

Orthocenter, Circumcenter (i.e., the options: Orthocenter, Circumcenter)