Where can the medians of a triangle intersect?

I. inside the triangle
II. on the triangle
III. outside the triangle
A. I only
B. III only
C. I or III only
D. I, II, or II

I only

The medians of a triangle can intersect inside the triangle, on the triangle, or outside the triangle. Therefore, the correct answer is D. I, II, or III.

To determine where the medians of a triangle intersect, let's first understand what medians are. The medians of a triangle are segments that connect each vertex of the triangle to the midpoint of the opposite side.

Now, let's consider the options:

I. Inside the triangle: When we draw the medians of a triangle, they always intersect at a point that lies inside the triangle. This point is called the centroid of the triangle, and it is the center of mass of the triangle.

II. On the triangle: The medians of a triangle can also intersect at a point that lies on one of the sides of the triangle, but not the vertex. This can occur in an isosceles or equilateral triangle, where the centroid coincides with one of the vertices.

III. Outside the triangle: The medians of a triangle do not intersect outside the triangle.

Based on the explanations above, the correct answer is (A) I only, as the medians can only intersect inside the triangle.