7
$\begingroup$

Suppose $a$, $b$, and $c$ are the lengths of the sides of a triangle, and $R$ and $r$ are its circumradius and inradius respectively. How can one prove the following inequality? $$2r^2+8Rr \leq \frac{a^2+b^2+c^2}{2}$$

  • 0
    I have modified the problem. In this way it should be easier to be proven. I don't have the solution. The image is irrelevant to the problem. I will delete it.2011-12-31
  • 8
    You, now, need to tell us what your symbols represent.2011-12-31
  • 1
    It is well known - R and r are the circumradius and inradius. a,b,c are the sides of the triangle.2011-12-31
  • 0
    If this is taken from a math competition, [tag:contest-math] tag would be appropriate.2011-12-31
  • 0
    Not really. Proved it using vectors but was looking for a simple proof. Well done.2011-12-31
  • 3
    @Chun-Yue Could you post your vector proof, please? I would like to see it.2011-12-31
  • 0
    Two suggestions were already made (to add the tag (contest-math) and to post your own proof), which you chose to ignore. So... let me add a third one: accept an answer (or explain why none satisfies you).2012-02-12

3 Answers 3