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

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The discussion centers on the geometric proof that the three medians of a triangle divide it into six similar triangles of equal area. Participants explore the properties of triangle medians, specifically how points D, E, and F, which lie on the sides of triangle ABC, create equal areas when intersected by the medians at point O. The conversation emphasizes the need for a mathematical proof rather than accepting claims without verification. The importance of understanding the relationship between the areas of the triangles formed by the medians is highlighted. Ultimately, the discussion seeks clarity on how the median divides the triangle's area into equal portions.
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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.
 
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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...
 
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
 
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.
 
the base and the height of those two triangles have equal value
 
Well,Vincentchan,i was obviously asking the OP...

Daniel.

PS.I knew you could answer it... :-p
 

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