本帖最后由 xzlbq 于 2017-12-12 10:26 编辑
在“珠峰不等式“( 微信号zhufengbudengshi)第64期中,我把这个不等式推广为三角形规范几何量不等式,即有不等式
\[\left( b+c \right) {\it h_a}+{\it h_c}\, \left( a+b \right) \geq b{\it h_b}+
2\sqrt {3}{\it h_a}{\it h_c}\]
但它等价于
\[{\frac {a+b}{c}}+{\frac {b+c}{a}}\geq 1+2\,\sqrt {3}\sin \left( B \right) \]
这个不等式又有加强式
\[{\frac {a+b}{c}}+{\frac {b+c}{a}}\geq 1+2(\sum{\tan{\frac{A}{2}}})\sin \left( B \right) \]
<=>
\[{\frac {y+z}{x+y}}+{\frac {z+x}{x+y}}+{\frac {z+x}{y+z}}+{\frac {x+y}{
y+z}}\geq 1+4\,{\frac {yz}{ \left( x+y \right) \left( y+z \right) }}+4\,{
\frac {xz}{ \left( x+y \right) \left( y+z \right) }}+4\,{\frac {xy}{
\left( x+y \right) \left( y+z \right) }}
\]
easy.Done! |