Interesting Geometry problem

[Hessam]

First of all note this is NOT HOMEWORK

This is an interesting geometry problem my old statistics and discrete math teacher gave to me, i solved it, however i just wanted to give it out..... very tough problem (at least for me, probably easy for most poeple here)

You have a triangle... the lengths of the three medians are
Ma = 3
Mb = 4
Mc = 5

so what is the area of the triangle?

good luck

[Kahsi]

Hi!

I'm not a native english speaking person so I don't get the "median" part. Do you mean lenght of the sides?

[matt grime]

No, he means the lengths of the medians which are the lines that pass from one corner to the mid point of the opposite sides. It isn't a particularly difficult question if you know how to describe the areas of triangles in terms of sides (at least on first inspection it seems there is an obvious argument to try).

[Kittel Knight]

...so what is the area of the triangle?

Let G be the barycenter of ABC, and let P be its symmetric point with respect to middle point of BC.

Then, the triangle BGP is formed by 2/3 of the medians (and it is rectangle).

So, BGParea = (2/3)^2 * (3*4)/2 = 8/3 .

But BGP is 1/3 of ABC.

So, ABCarea is 8.