Prove that in any acute angled triangle of sides a, b, c, semi perimeter p, in-radius r, and circumradius R, the following inequalities hold:
Let D be the foot of the altitude from A, and D will be on the side BC since the triangle has only acute angles. Now, by summing up and , we get
For the other part, we have the following equivalences:
—– call the above as relation I
But, and and by Minkowski’s inequality, we have . Then, we will have
which holds for all positive a and because of the AM-GM inequality. This tells us that (1) is true, and thus so is our conclusion.
Ref: Problems for the Mathematical Olympiads (from the First Team Selection Test to the IMO) by Andrei Negut
More geometry later !