Results 1 to 2 of 2

Thread: Triangle

  1. #1
    Senior Member
    Joined
    Nov 2007
    Posts
    329

    Triangle

    Prove that if the line through the centroid and the incenter is parallel to a side then the sides of the triangle form an arithmetic sequence.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor red_dog's Avatar
    Joined
    Jun 2007
    From
    Medgidia, Romania
    Posts
    1,252
    Thanks
    5
    Suppose that $\displaystyle IG\parallel BC$ (I=incenter, G=centroid).
    Let $\displaystyle r$ be the inradius, $\displaystyle h_a$ the height from the vertex A, $\displaystyle S$ the area of triangle and $\displaystyle p$ the semiperimeter.

    Then we have $\displaystyle r=\frac{h_a}{3}=\frac{2S}{3a}$

    But $\displaystyle r=\frac{S}{p}\Rightarrow\frac{2S}{3a}=\frac{S}{p}\ Rightarrow\frac{1}{3a}=\frac{1}{a+b+c}\Rightarrow a=\frac{b+c}{2}$ and that means that b, a, c (or c, a, b) are in arithmetic progression.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: Apr 23rd 2011, 08:10 AM
  2. Replies: 3
    Last Post: Apr 30th 2009, 07:41 AM
  3. Replies: 1
    Last Post: Oct 28th 2008, 07:02 PM
  4. Replies: 7
    Last Post: Jul 19th 2008, 06:53 AM
  5. Replies: 27
    Last Post: Apr 27th 2008, 10:36 AM

Search Tags


/mathhelpforum @mathhelpforum