Pedoe's inequality

The inequality of Pedoe or also inequality of Neuberg-Pedoe , named after Daniel Pedoe and Joseph Neuberg , is a geometric statement about the side lengths and the area of two triangles .

If a , b and c are the side lengths of a triangle with the area f and A , B and C are the side lengths of another triangle with the area F , then the following inequality applies:

${\displaystyle A^{2}(b^{2}+c^{2}-a^{2})+B^{2}(a^{2}+c^{2}-b^{2})+C^{2}(a^{2}+b^{2}-c^{2})\geq 16Ff,\,}$

The equals sign applies if the two triangles are similar to each other.

Note that the arithmetic expression on the left is not only symmetric with respect to the six permutations of the set {( A , a ), ( B , b ), ( C , c )} of ordered pairs , but also - perhaps less obviously - regarding the exchange of A with a , B with b and C with c . In other words, it is a symmetric function of the given pair of triangles.

This inequality generalizes Weitzenböck's inequality . This is obtained when one of the two triangles is equilateral, because then the side length of the equilateral triangle is reduced from the inequality and what remains is Weitzenböck's inequality for the second triangle.

Pedoe found the inequality in 1941 and published it in several articles. Later it turned out that the inequality had already been discovered by Neuberg in the 19th century, although Neuberg had not yet proven that the similarity of the two triangles follows from the equality.

literature

• Claudi Alsina, Roger B. Nelsen: When Less is More: Visualizing Basic Inequalities . MAA, 2009, ISBN 978-0-88385-342-9 , p. 108
• Gengzhe Chang, Thomas W. Sederberg: Over and Over Again . Cambridge University Press, 1997, ISBN 9780883856413 , pp. 74-75
• DS Mitrinović, Josip Pečarić: About the Neuberg-Pedoe and the Oppenheim inequalities . Journal of Mathematical Analysis and Applications 129 (1): 196-210 January 1988 ( online copy )
• Daniel Pedoe : An Inequality Connecting Any Two Triangles . The Mathematical Gazette, Vol. 25, No. 267 (Dec., 1941), pp. 310-311 ( JSTOR 3606570 )
• Daniel Pedoe: An Inequality for Two Triangles , In Proceedings of the Cambridge Philosophical Society, Vol. 38, Part 4, p. 397, 1943.
• Daniel Pedoe: A Two-Triangle Inequality , In The American Mathematical Monthly, Vol. 70, No. 9, p. 1012, November 1963.
• Daniel Pedoe: Thinking Geometrically . The American Mathematical Monthly, vol. 77, no. 7 (Aug - Sep, 1970), pp. 711-721 ( JSTOR 2316201 )
• Yang Lu, Zhang Jing-Zhong: A Generalization to Several Dimensions of the Neuberg-Pedoe Inequality, with Applications . BULL. AUSTRAL. MATH. SOC., VOL. 1983, 27: 203-214. ( Online copy )