I have been struggling with this problem, and would like to prove the inequality using the Cauchy-Schwarz Inequality:

The vertices of a fixed triangle are $A$,$B$ and $C$, and $P$,$Q$ and $R$ lie on the line segments $BC$, $CA$ and $AB$ respectively. if $[XYZ]$ denotes the area of the triangle $XYZ$, prove that enter image description here

Any help would be much appreciated!



It seems like we can just use AM-GM inequality to do it.

Let $m = CQ$, $n = QA$, $p = AR$, $q = RB$, $x = BP$, and $y = PC$. Also let $a = BC$, $b = AC$, and $c = AB$. Then:

$\sqrt{\dfrac{S_{AQR}}{S_{ABC}}} = \sqrt{\dfrac{np}{bc}} \leq \dfrac{1}{2}\cdot \left(\dfrac{n}{b} + \dfrac{p}{c}\right)$.

Similarly, we can obtain inequalities for the other two square-roots. Thus the sum $T$ of the left side satisfies:

$T \leq \dfrac{1}{2}\cdot \left(\dfrac{n}{b} + \dfrac{p}{c}\right) + \dfrac{1}{2}\cdot \left(\dfrac{q}{c} + \dfrac{x}{a}\right) + \dfrac{1}{2}\cdot \left(\dfrac{m}{b} + \dfrac{y}{a}\right) = \dfrac{3}{2}$. Done.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.