Abstract
Leta 1,b 1,c 1,A 1 anda 2,b 2,c 2,A 2 be the sides and areas of two triangles. Ifa=(a p1 +a p2 )1/p,b=(b p1 +b p2 )1/p,c=(c p1 +c p2 )1/p, and 1≤p≤4, thena, b, c are the sides of a triangle and its area satisfiesA p/2≥A p/21 +A p/22 . If obtuse triangles are excluded,p>4 is allowed. For convex cyclic quadrilaterals, a similar inequality holds. Also, leta, b, c, A be the sides and area of an acute or right triangle. Iff(x) satisfies certain conditions,f(a),f(b),f(c) are the sides of a triangle having areaA f, which satisfies (4A f/√3)1/2≥f((4A/√3)1/2).
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References
Bottema, O., Djordjević, R. Ž., Janić, R. R., Mitrinović, D. S. andVasić, P. M.,Geometric inequalities. Wolters-Noordhoff Publishing, Groningen, 1969, chapter 4.
Carroll, C. E., Yang, C. C. andAhn, S.,Some triangle inequalities and generalizations. Canad. Math. Bull.23 (1980), 267–274.
Jensen, J. L. W. V., Sur les fonctions convexes et les inégalitiés entre les valeurs moyennes. Acta Math.30 (1905), 175–193.
Minkowski, Herman,Geometrie der Zahlen. Chelsea Publishing Company, New York, 1953.
Oppenheim, A.,Some inequalities for triangles. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Math. Fiz.363 (1971), 21–28.
Oppenheim, A.,Inequalities involving elements of triangles, quadrilaterals or tetrahedra. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz.496 (1974), 257–263.
Polya, G.,Induction and analogy in mathematics. Princeton University Press, Princeton, 1954, pp. 176–177.
Steiner, J.,Gesammelte Werke. herausgegeben von K. Weierstrass, Chelsea Publishing Company, New York, 1971, vol. 2, p. 199.
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Carroll, C.E. Proof of Oppenheim's area inequalities for triangles and quadrilaterals. Aeq. Math. 24, 97–109 (1982). https://doi.org/10.1007/BF02193037
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DOI: https://doi.org/10.1007/BF02193037