Define the first Brocard point as the interior point 
of a triangle for which the angles 
, 
, and 
are equal to an angle 
. Similarly, define the second Brocard
point as the interior point 
for which the angles 
, 
, and 
are equal to an angle 
. Then 
, and this angle is called the Brocard angle.
The Brocard angle 
of a triangle 
is given by the formulas
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(1)
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(2)
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(3)
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(4)
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(5)
|
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(6)
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(7)
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(8)
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(9)
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(10)
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where 
is the triangle area, 
, 
,
and 
are angles, and 
, 
,
and 
are the side lengths (Johnson 1929). Equation (8) is due to Neuberg
(Tucker 1883).
Gallatly (1913, p. 96) defines the quantity 
as
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(11)
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If an angle 
of a triangle is given, the
maximum possible Brocard angle (and therefore minimum possible value of 
) is given by
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(12)
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(Johnson 1929, p. 289). If 
is specified, then the largest possible value 
and minimum possible value 
of any possible triangle having Brocard angle 
are given by
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(13)
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(14)
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where the square rooted quantity is the radius of the corresponding Neuberg circle (Johnson 1929, p. 288). The maximum possible Brocard angle (and therefore
minimum possible value of 
) for any triangle is 
(Honsberger 1995, pp. 102-103), so
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(15)
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The Abi-Khuzam inequality states that
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(16)
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(Abi-Khuzam 1974, Le Lionnais 1983), which can be used to prove the Yff conjecture that
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(17)
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(Abi-Khuzam 1974). Abi-Khuzam also proved that
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(18)
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Interestingly, (◇) is equivalent to
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(19)
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and (◇) is equivalent to
![2omega<=RadicalBox[{A, B, C}, 3],](/images/equations/BrocardAngle/NumberedEquation8.svg) |
(20)
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which are inequalities about the arithmetic and geometric mean, respectively.
