# Hexagon&Centroids

• Apr 9th 2009, 03:45 AM
Logic
Hexagon&Centroids
Greetings,

I have a lot of trouble mastering the vectors. Moreover, the teacher I am usually using for consult claims she is not the greatest expert in vectors, either, so I have few options left to seek out help.

We are given a hexagon as shown in the picture below and K, P, L, Q, M, R are the midpoints of AB, BC, CD, DE, EF and FA respectively. Proove that the triangles KLM and PQR have the same median point.

It is suggested in the textbook that if G1 and G2 are the median points of KLM and PQR to proove that vector OG1 equals vector OG2 for any point O.

Now, I know that OG1 for example would equal one third of the three vectors with beginning O and ends K, L and M. The same goes for G2 and P, Q and R.
However, I cannot find a link between, say, vector OK and vector OP to proove they are equal and finally get to what is suggested in the textbook.

Thank you.
• Apr 10th 2009, 10:50 AM
Hello Logic
Quote:

Originally Posted by Logic
We are given a hexagon as shown in the picture below and K, P, L, Q, M, R are the midpoints of AB, BC, CD, DE, EF and FA respectively. Proove that the triangles KLM and PQR have the same median point.

It is suggested in the textbook that if G1 and G2 are the median points of KLM and PQR to proove that vector OG1 equals vector OG2 for any point O.

Now, I know that OG1 for example would equal one third of the three vectors with beginning O and ends K, L and M. The same goes for G2 and P, Q and R.

You're right!

Now look at how the points K, L, and M are related to the vertices of the hexagon. For instance, K is the mid-point of AB. So $\vec{OK} = \tfrac12(\vec{OA} + \vec{OB})$.

Do the same for L and M.

Now add the three vectors $\vec{OK}, \vec{OL}, \vec{OM}$ together, using the expressions you've just written down, and divide by 3 to get $\vec{OG_1}$.

Repeat for the points P, Q and R and you're done!

Can you complete it now?