TOPICS
Brocard Angle
Define the firstBrocard point as the interior point
of atriangle for which theangles
,
, and
are equal to an angle
. Similarly, define the secondBrocard point as the interior point
for which theangles
,
, and
are equal to an angle
. Then
, and this angle is called the Brocard angle.
The Brocard angle
of atriangle
is given by the formulas
 |  |  | (1) |
 |  |  | (2) |
 |  |  | (3) |
 |  |  | (4) |
 |  |  | (5) |
 |  |  | (6) |
 |  |  | (7) |
 |  |  | (8) |
 |  |  | (9) |
 |  |  | (10) |
where
is thetriangle area,
,
, and
areangles, 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 anangle
of atriangle 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 correspondingNeuberg 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) |
TheAbi-Khuzam inequality states that
 | (16) |
(Abi-Khuzam 1974, Le Lionnais 1983), which can be used to prove theYffconjecture 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],](/image.pl?url=https%3a%2f%2fmathworld.wolfram.com%2fimages%2fequations%2fBrocardAngle%2fNumberedEquation8.svg&f=jpg&w=240) | (20) |
which are inequalities about the arithmetic and geometric mean, respectively.
See also
Abi-Khuzam Inequality,Brocard Circle,Brocard Line,Brocard Triangles,Equi-Brocard Center,Fermat Points,Neuberg Circles,Yff ConjectureExplore with Wolfram|Alpha
More things to try:
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. Monthly96, 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. Dublin: Hodges, Figgis, & Co., p. 172, 1888.Coolidge, J. L.A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 61, 1971.Emmerich, A.Die Brocardschen Gebilde und ihre Beziehungen zu den verwandten merkwürdigen Punkten und Kreisen des Dreiecks. Berlin: Reimer, 1891.Gallatly, W.The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 95, 1913.Honsberger, R. "The Brocard Angle." §10.2 inEpisodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 101-106, 1995.Johnson, R. A.Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 263-286 and 289-294, 1929.Lachlan, R.An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 65-66, 1893.Le Lionnais, F.Les nombres remarquables. Paris: Hermann, p. 28, 1983.Tucker, R. "The 'Triplicate Ratio' Circle."Quart. J. Pure Appl. Math.19, 342-348, 1883.Referenced on Wolfram|Alpha
Brocard AngleCite this as:
Weisstein, Eric W. "Brocard Angle." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/BrocardAngle.html