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 ABCDEF (namely A'B' and D'E', 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.
Note: I looked at https://web2.0calc.com/questions/please-help_7172 and https://web2.0calc.com/questions/geometry-polygons but I couldn't really understand anyone's solutions.
You can use coordinates: Let A = (a_1, a_2), etc. Then the rest is straight-forward calculations.
The hexagon isn't regular, so if I gave A a coordinate value of (0, 0), I still wouldn't know what the coordinates of other points would be. Here's the asymptote code:
[asy]
unitsize(1 cm);
pair[] A, B, C, D, E, F;
A[0] = (3,6);
B[0] = (5,5);
C[0] = (4,2);
D[0] = (0,-1);
E[0] = (-2,2);
F[0] = (0,5);
B[1] = (A[0] + B[0] + C[0])/3;
C[1] = (B[0] + C[0] + D[0])/3;
D[1] = (C[0] + D[0] + E[0])/3;
E[1] = (D[0] + E[0] + F[0])/3;
F[1] = (E[0] + F[0] + A[0])/3;
A[1] = (F[0] + A[0] + B[0])/3;
draw(A[0]--B[0]--C[0]--D[0]--E[0]--F[0]--cycle);
draw(A[1]--B[1]--C[1]--D[1]--E[1]--F[1]--cycle);
label("$A$", A[0], N);
label("$B$", B[0], dir(0));
label("$C$", C[0], SE);
label("$D$", D[0], S);
label("$E$", E[0], W);
label("$F$", F[0], NW);
label("$A'$", A[1], N);
label("$B'$", B[1], NE);
label("$C'$", C[1], SE);
label("$D'$", D[1], S);
label("$E'$", E[1], SW);
label("$F'$", F[1], NW);
[/asy]
Thanks for your help!
