Equality occurs if and only if the triangle is equilateral. Pedoe's inequality is a generalization of Weitzenböck's inequality. The Hadwiger-Finsler inequality is a strengthened version of Weitzenböck's inequality.
The proof of this inequality was set as a question in the International Mathematical Olympiad of 1961. Even so, the result is not too difficult to derive using Heron's formula for the area of a triangle:
This method assumes no knowledge of inequalities except that all squares are nonnegative.
and the result follows immediately by taking the positive square root of both sides. From the first inequality we can also see that equality occurs only when and the triangle is equilateral.
As we have used the rearrangement inequality and the arithmetic-geometric mean inequality, equality only occurs when and the triangle is equilateral.
so the expression in parentheses must be greater than or equal to 0.
References & further reading
- Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA 2010, ISBN 9780883853481, pp. 96–98 (excerpt (Google), p. 96, at Google Books)
- D. M. Batinetu-Giurgiu, Nicusor Minculete, Nevulai Stanciu: Some geometric inequalities of Ionescu-Weitzebböck type. International Journal of Geometry, Vol. 2 (2013), No. 1, April
- Claudi Alsina, Roger B. Nelsen: Geometric Proofs of the Weitzenböck and Hadwiger-Finsler Inequalities. Mathematics Magazine, Vol. 81, No. 3 (Jun., 2008), pp. 216–219 (JSTOR)
- Coxeter, H.S.M., and Greitzer, Samuel L. Geometry Revisited, page 64.