Answer:

To prove that the centroid of triangle ABC and triangle DEF are the same point, you can use the properties of medians in triangles.

First, recall that the centroid of a triangle is the point where all three medians intersect. In other words, AD, BE, and CF intersect at the centroid of triangle ABC.

Now, consider triangle DEF, which is formed by the medians of triangle ABC.

1. DE is parallel to BC, as DE is a median of triangle ABC, and BC is also a median of the same triangle.

2. Similarly, EF is parallel to AC, and DF is parallel to AB.

Now, let's prove that DE, EF, and DF are medians of triangle DEF:

a. DE is a median of triangle DEF because it connects vertex D (opposite to side EF) to the midpoint of EF.

b. EF is a median of triangle DEF because it connects vertex E (opposite to side DF) to the midpoint of DF.

c. DF is a median of triangle DEF because it connects vertex F (opposite to side DE) to the midpoint of DE.

Since DE, EF, and DF are medians of triangle DEF, they must intersect at the centroid of triangle DEF.

Now, since DE, EF, and DF are corresponding medians of triangles ABC and DEF, and they all intersect at the same point (the centroid of triangle ABC), it follows that the centroid of triangle ABC and triangle DEF are indeed the same point.

This proves that the centroids of triangle ABC and triangle DEF coincide.