Let and be the lengths of the sides of a nondegenerate triangle, let , and let and be the inradius and circumradius of the triangle, respectively. Show that

and determine the cases of equality.

**My solution.**

*A)Proof of the 1st inequality:*

Assume that is area of the triangle, then and , therefore that inequality is equivalent to

Setting , then and we need only prove

By AM-GM we have and therefore and also 1st inequality proved.

Equality occur iff iff triangle is equilateral.

*B)Proof of the 2nd inequality:*

By AM-GM we have

and we’re done. Equality occur iff iff .

Link download http://tuan.nguyentrung.googlepages.com/AMM_11306.pdf

### Like this:

Like Loading...

*Related*