1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove that a point in triangle is the centroid

  1. Feb 13, 2015 #1
    • Member warned about posting with no template
    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. jcsd
  3. Feb 13, 2015 #2

    wabbit

    User Avatar
    Gold Member

    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?
     
  4. Feb 13, 2015 #3

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

  5. Feb 13, 2015 #4

    wabbit

    User Avatar
    Gold Member

    I'd say don't even look at that, or you're gonna write pages of algebra and not understand what's happening. :)
     
  6. Feb 13, 2015 #5
    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.
     
  7. Feb 13, 2015 #6

    wabbit

    User Avatar
    Gold Member

    Aha. Indeed you do . What if there were two different points with the equal area property?
     
  8. Feb 14, 2015 #7

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Prove that a point in triangle is the centroid
Loading...