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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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