In the above trapezoid, we observe that \(ax\geq br+cq\). Similarly, we have \(by\geq ar+cp\ and\ cz\geq aq+bp\). Adding up the above inequalities yield
\begin{equation} x+y+z\geq\left(\frac{b}{a}+\frac{a}{b}\right)r+\left(\frac{c}{a}+\frac{a}{c}\right)q+(\frac{c}{b}+\frac{b}{c})p\nonumber \\ \end{equation}
By using the well known AM\(\geq\text{QM}\) , the Erdös-Mordell inequality is proved. If we apply the AM\(\geq QM\) again to the Erdös-Mordell inequality, we will derive that \(xyz\geq 8pqr\).
48. Example 7.3 Prove the Inequality
needs.
Proof: Draw lines OA,OB,OC, the point O, that | OA|=a, |OB|=b, |OC|=c, ∠AOB=\(\frac{\pi}{3}\) ,∠BOC = \(\frac{\pi}{3}\), ∠AOC=\(\frac{2\pi}{3}\) .