Question

where do the points of concurrency of certain types of triangles lie?
the circumcenter of an obtuse triangle lies
the triangle.
the incenter of a right triangle lies
the triangle.
the point of concurrency of the angle bisectors of an acute triangle lies
the triangle.
the point of concurrency that is equidistant from the vertices of a right triangle lies
the triangle.

Explanation:

1. For the circumcenter of an obtuse triangle: The circumcenter is the intersection of perpendicular bisectors. In an obtuse triangle, it lies outside the triangle.
2. For the incenter of a right triangle: The incenter is the intersection of angle bisectors and is always inside the triangle, regardless of the triangle type (acute, right, obtuse).
3. For the point of concurrency of angle bisectors (incenter) of an acute triangle: The incenter is inside any triangle, including acute triangles.
4. For the point equidistant from vertices (circumcenter) of a right triangle: In a right triangle, the circumcenter is at the midpoint of the hypotenuse, which is on the triangle.

Answer:

1. The circumcenter of an obtuse triangle lies outside the triangle.
2. The incenter of a right triangle lies inside the triangle.
3. The point of concurrency of the angle bisectors of an acute triangle lies inside the triangle.
4. The point of concurrency that is equidistant from the vertices of a right triangle lies on the triangle.