Prove that area of an arbitrary triangle is less than one fourth of the square of its perimeter.
(The question is from my sequences and series text)
Hello fardeen_genDo you know the formula for the area of a triangle: $\displaystyle \Delta = \sqrt{s(s-a)(s-b)(s-c)}$, where the semi-perimeter $\displaystyle s = \tfrac12(a+b+c)$?
Well, each of $\displaystyle (s-a), (s-b), (s-c)$ is less than $\displaystyle s$. So $\displaystyle \Delta < \sqrt{s^4} = \tfrac14(a+b+c)^2$.
Grandad