Weizenbock’s inequality
In a triangle with sides , , , and with area , the following inequality holds:
The proof goes like this: if is the semiperimeter of thetriangle, then from Heron’s formula 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 equivalent to:
Equality is achieved if and only if (i.e. when the triangle is equilateral) .
See also the Hadwiger-Finsler inequality, from which this result follows as a corollary.