Problem:

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:

Proof:

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 !

Nalin Pithwa