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Yff Conjecture

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The conjecture that, for any triangle,

8omega^3<ABC
(1)

where A, B, and C are the vertex angles of the triangle and omega is the Brocard angle. The Abi-Khuzam inequality states that

sinAsinBsinC<=((3sqrt(3))/(2pi))^3ABC
(2)

(Yff 1963, Le Lionnais 1983, Abi-Khuzam and Boghossian 1989), which can be used to prove the conjecture (Abi-Khuzam 1974).

The maximum value of ABC-8omega^3 occurs when two angles are equal, so taking A=B, and using A+B+C=pi, the maximum occurs at the maximum of

f(A)=A^2(pi-2A)-8{cot^(-1)[2cotA-cot(2A)]}^3,
(3)

which occurs when

2A(pi-3A) -(48{cot^(-1)[1/2(3cotA+tanA)]}^2[1+2cos(2A)])/(5+4cos(2A))=0.
(4)

Solving numerically gives A approx 1.40936 (OEIS A133844), corresponding to a maximum value of approximately 0.440053 (OEIS A133845).

See also

Abi-Khuzam Inequality, Brocard Angle

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References

Abi-Khuzam, F. "Proof of Yff's Conjecture on the Brocard Angle of a Triangle." Elem. Math. 29, 141-142, 1974.Abi-Khuzam, F. F. and Boghossian, A. B. "Some Recent Geometric Inequalities." Amer. Math. Monthly 96, 576-589, 1989.Bottema, O. "On Yff's Inequality for the Brocard Angle Triangle." Elem. Math. 31, 13-14, 1979.Klamkin, M. S. "On Yff's Inequality for the Brocard Angle of a Triangle." Elem. Math. 32, 188, 1977.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.Sloane, N. J. A. Sequences A133844 and A133845 in "The On-Line Encyclopedia of Integer Sequences."Yff, P. "An Analog of the Brocard Points." Amer. Math. Monthly 70, 495-501, 1963.

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Yff Conjecture

Cite this as:

Weisstein, Eric W. "Yff Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/YffConjecture.html

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