请输入您要查询的字词:

 

单词 WeizenbocksInequality
释义

Weizenbock’s inequality


In a triangleMathworldPlanetmath with sides a, b, c, and with area A, the following inequality holds:

a2+b2+c24A3

The proof goes like this: if s=a+b+c2 is the semiperimeter of thetriangle, then from Heron’s formulaMathworldPlanetmathPlanetmath we have:

A=s(s-a)(s-b)(s-c)

But by squaring the latter and expanding the parentheses we obtain:

16A2=2(a2b2+a2c2+b2c2)-(a4+b4+c4)

Thus, we only have to prove that:

(a2+b2+c2)23[2(a2b2+a2c2+b2c2)-(a4+b4+c4)]

or equivalently:

4(a4+b4+c4)4(a2b2+a2c2+b2c2)

which is trivially equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to:

(a2-b2)2+(a2-c2)2+(b2-c2)20

Equality is achieved if and only if a=b=c (i.e. when the triangle is equilateral) .

See also the Hadwiger-Finsler inequality, from which this result follows as a corollary.

随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 19:29:44