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
![2omega<=RadicalBox[{A, B, C}, 3],](https://mathworld.wolfram.com/images/equations/BrocardAngle/NumberedEquation8.svg) | (20) |
which are inequalities about the arithmetic and geometric mean, respectively.
