Weizenbock’s inequality

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


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


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


Thus, we only have to prove that:


or equivalently:


which is trivially equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to:


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.

Title Weizenbock’s inequality
Canonical name WeizenbocksInequality
Date of creation 2013-03-22 13:19:33
Last modified on 2013-03-22 13:19:33
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Theorem
Classification msc 51F99
Related topic HadwigerFinslerInequality