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Upper and Lower Bounds for The Area of a Triangle for Whose Sides Two Symmetric Functions are Known

Published online by Cambridge University Press:  20 November 2018

Robert Frucht
Robert Frucht*
Affiliation:
Universidad Técnica F. Santa Maria Valparaiso, Chile
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Improving on inequalities given by Gerretsen (2), Beatty (1) has proved that for the area Δ of any plane triangle with sides a, b, c the following inequalities hold:

(1.1)

where

(1.2);

the signs of equality in (1.1) only apply when the triangle is equilateral. Beatty has also remarked that the second inequality in (1.1) is of no value in case 5H ≥ 3K, since then the lower estimate which it gives for Δ2 is not even positive.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 9 , 1957 , pp. 227 - 231
Copyright
Copyright © Canadian Mathematical Society 1957

References

1. Beatty, S., Upper and lower estimates for the area of a triangle, Trans. Roy. Soc. Canada, III(3), 48 (1954), 1–5.Google Scholar
2. Gerretsen, J. C. H., Ongelijkheden in de driehoek, Nieuw Tijdschr. Wiskunde, 41 (1953), 1–7.Google Scholar
3. Griffiths, Lois W., Introduction to the Theory of Equations (2nd ed., New York, 1947).Google Scholar