Prove that a point in triangle is the centroid

[Purplesquiggles]

You are given an arbitrary triangle ABC. Inside ABC there is a point M such that Area(ABM) = Area(BCM) = Area(ACM) . Prove that M is the centroid of triangle ABC.

I have had very little progress with this question. I've tried connecting a line from M which bisects BC, but I cannot prove that the two lines are collinear.

I've also tried continuing on AM until it intersects BC., but I cannot prove it bisects BC.

Does anyone have any ideas?

 

[wabbit]

Sometimes it helps trying to do it the other way. Say Q is the centroid. Can you prove that the areas ABQ BCQ and CAQ are the same?

 

[PeroK]

This may be useful:

http://en.wikipedia.org/wiki/Heron's_formula

 

[wabbit]

I'd say don't even look at that, or you're going to write pages of algebra and not understand what's happening. :)

 

[Purplesquiggles]

Sometimes it helps trying to do it the other way. Say Q is the centroid. Can you prove that the areas ABQ BCQ and CAQ are the same?

I can prove what you said quite easily. The only problem is that the converse, which is my initial question, remains unproved. I don't know for certain that 3 triangles of equal area must meet at the centroid. Its possible there are other places where this can occur. I have to prove the centroid is the only one.

 

[wabbit]

Aha. Indeed you do . What if there were two different points with the equal area property?

 

[vela]

Can you show that all points X such that triangles ABX and ACX have the same area lie on a line? If so, this is the line that connects A to the midpoint of BC.