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
(1)
| |||
(2)
| |||
(3)
| |||
(4)
| |||
(5)
| |||
(6)
| |||
(7)
| |||
(8)
| |||
(9)
| |||
(10)
|
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
(11)
|
If an angle of a triangle is given, the maximum possible Brocard angle (and therefore minimum possible value of ) is given by
(12)
|
(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
(13)
| |||
(14)
|
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
(15)
|
The Abi-Khuzam inequality states that
(16)
|
(Abi-Khuzam 1974, Le Lionnais 1983), which can be used to prove the Yff conjecture that
(17)
|
(Abi-Khuzam 1974). Abi-Khuzam also proved that
(18)
|
Interestingly, (◇) is equivalent to
(19)
|
and (◇) is equivalent to
(20)
|
which are inequalities about the arithmetic and geometric mean, respectively.