# Prove that a point in triangle is the centroid

1. Feb 13, 2015

### Purplesquiggles

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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?

2. Feb 13, 2015

### 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?

3. Feb 13, 2015

### PeroK

4. Feb 13, 2015

### wabbit

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

5. Feb 13, 2015

### Purplesquiggles

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.

6. Feb 13, 2015

### wabbit

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

7. Feb 14, 2015

### vela

Staff Emeritus
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.