Brocard Angle


Define the first Brocard point as the interior point Omega of a triangle for which the angles ∠OmegaAB, ∠OmegaBC, and ∠OmegaCA are equal to an angle omega. Similarly, define the second Brocard point as the interior point Omega^' for which the angles ∠Omega^'AC, ∠Omega^'CB, and ∠Omega^'BA are equal to an angle omega^'. Then omega=omega^', and this angle is called the Brocard angle.

The Brocard angle omega of a triangle DeltaABC is given by the formulas


where Delta is the triangle area, A, B, and C are angles, and a, b, and c are the side lengths (Johnson 1929). Equation (8) is due to Neuberg (Tucker 1883).

Gallatly (1913, p. 96) defines the quantity e as


If an angle alpha of a triangle is given, the maximum possible Brocard angle (and therefore minimum possible value of cotomega) is given by


(Johnson 1929, p. 289). If omega is specified, then the largest possible value alpha_(max) and minimum possible value alpha_(min) of any possible triangle having Brocard angle omega are given by


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 cotomega) for any triangle is 30 degrees (Honsberger 1995, pp. 102-103), so


The Abi-Khuzam inequality states that


(Abi-Khuzam 1974, Le Lionnais 1983), which can be used to prove the Yff conjecture that


(Abi-Khuzam 1974). Abi-Khuzam also proved that


Interestingly, (◇) is equivalent to


and (◇) is equivalent to

 2omega<=RadicalBox[{A, B, C}, 3],

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 Conjecture

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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. 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 in Episodes 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.

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Brocard Angle

Cite this as:

Weisstein, Eric W. "Brocard Angle." From MathWorld--A Wolfram Web Resource.

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