triangle property?

Question

Question 1
--------------------
is there any rule that For any triangle orthocentre (H) , centroid (G), circumcentre(C) are Collinear ?


Question 2
------------------
does G divides HC in 2:1 ? i know G divides its median in 2:1 but does it do for HC also ?

Answer 1

1) Yes there's a proof but it's hard to show here. The line containing them is called the Euler's line.

You can see that here:

http://new.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8B%E0%A4%A3

2) I'm not sure about that either...

Answer 2

Yes there is a rule. The line is indeed called the Euler line.

G does divide HC in 2:1 ratio. However it is a different property altogether from the fact that it also divides the median.

See this site for a proof of the theorem.
http://www.ies.co.jp/math/java/vector/veuler/veuler.html

Incidentally the nine point circle center also lies on the euler line.

Answer 3

The answer is yes to both questions. To see that draw the parallel to each side passing through the opposing vertex. You get a new triangle which is twice bigger than the original one. Now H is the circumcenter of the big triangle. So the dilation of center G and ratio (-2) which sends the small triangle to the big one also sends O to H. There you are.