Name all types of triangles for which the point of concurrency is inside the triangle?

Answer 1

The answer depends on what point of concurrency you are referring to. There are four segments you could be talking about in triangles. They intersect in different places in different triangles.

Medians--segments from a vertex to the midpoint of the opposite side. In acute, right and obtuse triangles, the point of concurrency of the medians (centroid) is inside the triangle.

Altitudes--perpendicular segments from a vertex to a line containing the opposite side. In an acute triangle, the point of concurrency of the altitudes (orthocenter) is inside the triangle, in a right triangle it is on the triangle and in an obtuse triangle it is outside the triangle.

Perpendicular bisectors of sides--segments perpendicular to each side of the triangle that bisect each side. In an acute triangle, the point of concurrency of the perpendicular bisectors (circumcenter) is inside the triangle, in a right triangle it is on the triangle and in an obtuse triangle it is outside the triangle.

Angle bisectors--segments from a vertex to the opposite side that bisect the angles at the vertices. In acute, right and obtuse triangles, the point of concurrency of the angle bisectors (incenter) is inside the triangle.