## Thursday, December 10, 2009

### Problem 402. Hexagon, Triangle, Centroid, Parallel, Congruence, Similarity

Proposed Problem
Click the figure below to see the complete problem 402 about Hexagon, Triangle, Centroid, Parallel, Congruence, Similarity. Complete Problem 402
Level: High School, SAT Prep, College geometry

1. My problems for this picture.
Proff
1)Area hexagon ABCDEF/ Area hexagon GHJKLM = 9/4.

2) Area polygon ABCDE/ Area polygon GHJKL = 117/32.

2) Area polygon ABCD/ Area polygon GHJK = 9.

2. the affix of triangle ABC centroid G is
g=(a+b+c)/3
then
3(h-g)= b+c+d-a-b-c= d-a
3(k-l)= d+e+f-e+f+a= d-a
this means that GH=LK and GH // LK
.........

3. We see that
XH=(XB+XC+XD)/3
XG=(XB+XC+XA)/3
XK=(XD+XE+XF)/3
XL=(XE+XF+XA)/3
and HG=(XH-XG,YH-YG)=(XD-XA,YD-YA)/3
LK=(XK-XL,YK-YL)=(XD-XA,YD-YA)/3
etc.
And my question for this picture - prove
area hexagon ABCDEF/area hexagon GHJKLM = 9/4

4. Interesting about the areas, but the result about the formed hexagon having opposite sides equal & parallel can be generalized for 2n-gons as shown here: http://dynamicmathematicslearning.com/octagoncentroids.html