Why do the 3 medians of a triangle create 6 similar triangles with equal area?

[Perplexing]

I know that the 3 medians of a triangle split the triangle into 6 similair triangles with equal area.

I wish to know the proof for this using simple geometry.

 

[dextercioby]

How do you know that??Did someone tell u,or you just invented it??

Daniel.

PS.If someone did tell u,why didn't u ask for the proof??Did u buy it,just like that??In mathematics,things don't work that way...

 

[vincentchan]

let's say...
you have a triangle ABC...
D is the point in between A and B
E is the point in between B and C
F is the point in between C and A
and O is the point 3 median lines intersect each other...

AD=DB,AF=FC,BE=EC by definition
area ADO=area BDO, area AFO=CFO, area BEO = CEO
area ABE=ACE

area ABO=ABE-BEO
area ACO=ACE-CEO
this implies ABO=ACO
since ADO=DBO, ADO=1/2ABO
since AFO=FCO, AFO= 1/2AFO=1/2ABO=ADO,
so, AFO=FCO=ABO=ADO
use the same method, you can show AFP=FCO=ABO=ADO=BEO=CEO

 

[dextercioby]

So i think everything comes down to proving that the median of a side of a triangle divides the area enclosed by the triangle into 2 equisuperficial portions...

Can u show that??

BTW,the triangle does not have an area...

Daniel.

 

[vincentchan]

the base and the height of those two triangles have equal value

 

[dextercioby]

Well,Vincentchan,i was obviously asking the OP...

Daniel.

PS.I knew you could answer it...