Final answer:
By utilizing the properties of centroids and their relations to the sides and vertices of a triangle, we can prove that opposite sides of the hexagon formed by centroids are parallel and equal in length, and that the triangles formed by connecting alternate vertices of this hexagon have equal areas.
Step-by-step explanation:
The problem presented involves a convex hexagon abcdef and its associated centroids in each of its triangles. The key to solving part (a) lies in understanding the properties of centroids in a triangle, which is that they are the points of concurrency of the medians and divide each median in a 2:1 ratio, with the longer segment being adjacent to the vertex of the triangle. Due to this, if the triangles fab, abc, bcd, cde, def, efa have congruent corresponding sides, their centroids lie at the same proportion from the vertices, creating parallel lines on opposite sides of the new hexagon a'b'c'd'e'f'. Since the corresponding sides of the original hexagon are equal (due to it being convex and the definition of the centroids), the opposite sides of the new hexagon created by connecting the centroids will also be equal and parallel. For part (b), the knowledge about the area of triangles and the properties of centroids is used. If opposite sides of the hexagon a'b'c'd'e'f' are equal and parallel, it creates a scenario where triangles a'c'e' and b'd'f' not only share base-length relationships but also height relationships, thus making their areas equal.