## Solution of problem 11306 in AMM

Let $a,b,$ and $c$ be the lengths of the sides of a nondegenerate triangle, let $p=(a+b+c)/2$, and let $r$ and $R$ be the inradius and circumradius of the triangle, respectively. Show that $\dfrac{a}{2}\cdot \dfrac{4r-R}{R}\leq\sqrt{(p-b)(p-c)}\leq \dfrac{a}{2},$

and determine the cases of equality.

My solution.

## Problems in Section 1 of GTM 167

Please post carefully solutions of the following ones:

1. Let $K$ be a field extension of $F$. By defining scalar multiplication for $\alpha\in F$ and $a\in K$ by $\alpha\cdot a=\alpha a$, the multiplication in $K$, show that $K$ is an $F-$ vector space.

2. If $K$ is a field extention of $F$, prove that $[K:F]=1$ iff $K=F$.