The Brocard angle triangle angle for which
first Brocard point . The second
Brocard point angle , with
The Brocard angle triangle
where triangle area , angles , and side lengths (Johnson 1929). Equation
(8 ) is due to Neuberg (Tucker 1883).
Morel (1883, §23, p. 169) gives a geometric construction of the Brocard angle using a straightedge
and compass . Draw the tangent
line to the circumcircle of intersection with
the line through parallel to
Gallatly (1913, p. 96) defines the quantity
(11)
If an angle triangle is given, the
maximum possible Brocard angle (and therefore minimum possible value of
(12)
(Johnson 1929, p. 289). If triangle
having Brocard angle
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 triangle
is
(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, (15 ) is equivalent to
(19)
and (18 ) is equivalent to
(20)
which are inequalities about the arithmetic
mean and geometric mean , respectively.
See also Abi-Khuzam Inequality ,
Brocard Circle ,
Brocard
Line ,
Brocard Triangles ,
Equi-Brocard
Center ,
Fermat Points ,
First
Brocard Triangle ,
Neuberg Circles ,
Yff
Conjecture
Explore with Wolfram|Alpha
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. Casey, J. A
Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction
to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Coolidge, J. L. A
Treatise on the Geometry of the Circle and Sphere. Emmerich, A. Die Brocardschen Gebilde und ihre Beziehungen zu
den verwandten merkwürdigen Punkten und Kreisen des Dreiecks. Berlin, Germany:
Reimer, 1891. Gallatly, W. The
Modern Geometry of the Triangle, 2nd ed. Honsberger, R. "The Brocard Angle." §10.2 in Episodes
in Nineteenth and Twentieth Century Euclidean Geometry. Johnson, R. A. Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Lachlan,
R. An
Elementary Treatise on Modern Pure Geometry. Le Lionnais, F. Les
nombres remarquables. Morel,
A. "Étude sur le cercle de Brocard." Journal de mathématiques
élémentaires (2) 2 , 10-15, 33-37, 62-66, 97-103, 169-176,
and 195-201, 1883. https://archive.org/details/s2journaldemathm02pari . Tucker,
R. "The 'Triplicate Ratio' Circle." Quart. J. Pure Appl. Math. 19 ,
342-348, 1883. Referenced on Wolfram|Alpha Brocard Angle
Cite this as:
Weisstein, Eric W. "Brocard Angle." From
MathWorld https://mathworld.wolfram.com/BrocardAngle.html
Subject classifications
of a triangle 
is the common angle for which
,
where 
is the first Brocard point. The second
Brocard point 
gives the same angle, with 
.
of a triangle 
is given by the formulas
is the triangle area, 
, 
,
and 
are angles, and 
, 
,
and 
are the side lengths (Johnson 1929). Equation
(8) is due to Neuberg (Tucker 1883).
at 
, and let 
be its intersection with
the line through 
parallel to 
. Then 
.
as
of a triangle is given, the
maximum possible Brocard angle (and therefore minimum possible value of 
) is given by
is specified, then the largest possible value 
and minimum possible value 
of any possible triangle
having Brocard angle 
are given by
) for any triangle
is 
(Honsberger 1995, pp. 102-103),
so
![2omega<=RadicalBox[{A, B, C}, 3],](/images/equations/BrocardAngle/NumberedEquation8.svg)
