# 芬斯拉不等式

${\displaystyle a^{2}+b^{2}+c^{2}\geq 4{\sqrt {3}}S}$ （当且仅当${\displaystyle a=b=c}$时，等号成立）……（1）

${\displaystyle a^{2}=(m+n)^{2}}$${\displaystyle b^{2}=h^{2}+n^{2}}$${\displaystyle c^{2}=h^{2}+m^{2}}$${\displaystyle S={\frac {(m+n)h}{2}}}$
${\displaystyle [h-{\tfrac {{\sqrt {3}}(m+n)h}{2}}]^{2}+[{\tfrac {(m-n)h}{2}}]^{2}\geq 0}$ ……（2）

${\displaystyle (m+n)^{2}+h^{2}+n^{2}+h^{2}+m^{2}\geq 2{\sqrt {3}}(m+n)h}$
${\displaystyle a^{2}+b^{2}+c^{2}\geq 4{\sqrt {3}}S}$。（当${\displaystyle a=b=c}$时，等号成立）

${\displaystyle a^{2}+b^{2}+c^{2}-4{\sqrt {3}}S}$
${\displaystyle =a^{2}+b^{2}+(a^{2}+b^{2}-2ab\cos C)-2{\sqrt {3}}ab\sin C}$
${\displaystyle =2[a^{2}+b^{2}-2ab\sin }$(C+30°)
${\displaystyle \geq 2(a^{2}+b^{2}-2ab)=2(a-b)^{2}\geq 0}$

## 芬斯拉不等式的推广

1、若a、b、c、d为四边形的四条边，S为其面积，则有

${\displaystyle a^{2}+b^{2}+c^{2}+d^{2}\geq 4S}$

2、若${\displaystyle L_{1}}$${\displaystyle L_{2}}$、……、${\displaystyle L_{n}}$为n边形的边长，S为其面积，则有

${\displaystyle L_{1}^{2}+L_{2}^{2}+}$……${\displaystyle L_{n}^{2}\geq 4S\tan {\tfrac {\pi }{n}}(n\geq 3)}$