where ,
,
and
are the vertex angles of a triangle. The maximum is
reached for an equilateral triangle (and
therefore at )
and has numerical value 0.56559562463... (OEIS A127205).
The inequality was proven by Abi-Khuzam (1974), also considered by Klamkin (1977), and mentioned by Flanders (1978) as "an interesting related result" for
the product
of the three angles , , and of a triangle.
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. Monthly96, 576-589, 1989.Flanders, H. "Review
of 'Problems and Theorems in Analysis,' by Pólya and Szegö." Bull.
Amer. Math. Soc.86, 53-62, 1978.Klamkin, M. S. "On
Yff's Inequality for the Brocard Angle of a Triangle." Elem. Math.32,
188, 1977.Kuipers, L. "Extension of Abi-Khuzam's Inequality to
More than Four Angles." Nieuw Tijdschr. Wisk.69, 166-169, 1981-1982.Sloane,
N. J. A. Sequence A127205 in "The
On-Line Encyclopedia of Integer Sequences."
,
,
and 
are the vertex angles of a triangle. The maximum is
reached for an equilateral triangle (and
therefore at 
)
and has numerical value 0.56559562463... (OEIS A127205).
of the three angles 
, 
, and 
of a triangle.
