Let $ABCDEF$ be a convex hexagon.
Let A', B', C', D', E', F' be the centroids of triangles FAB, ABC, BCD, CDE, DEF, EFA, respectively.
(a) Show that every pair of opposite sides in hexagon A'B'C'D'E'F' (namely A'B' and D'E' and B'C' and E'F' and C'D'and F'A') are parallel and equal in length.
(b) Show that triangles A'C'E' and B'D'F' have equal areas.
Thank you for any help,
I think it would help a lot if you posted a diagram or a picture of your problem, especially if there is one in your textbook or assignment. If there isn't one, I recommend that you make an effort in drawing one to the best of your understanding and ability. It will also make it a lot easier for other people to see your problem in black and white, and they would be much more likely to attempt to solve it.
Here is a diagram. Not everything is included becasue it is too complex.
The maths is easy but the display is not.
You have to prove that the green triangle and the purple triangle in the middle have equal area.
To do this you have to look at the middle hexagon, the outside one is irrelevant.
You can prove that the three sides are congruent pairs and then you can say the triangles are congruent.
(3 equal sided test) so therefore the areas must be equal.
I'll do one side for you and leave you to the others.
Conside B'F' and E'C'
B'A=E'D' and they are also parallel given
A'F'=D'C' and they are also parallel given
angleB'A'F' = angleE'D'F' since their rays are parallel
therefore triangle A'B'F' congruent to triangle D'E'F' (side angle side test)
so B'F' = E'C'
By the same logic the other two corresponding sides will also be congruent
therefore the green and purple triangles are congruent (3 equal side test)
Since they are congruent triangles they must have the same area.