Brocard Points


The first Brocard point is the interior point Omega (also denoted tau_1 or Z_1) of a triangle DeltaABC with points labeled in counterclockwise order for which the angles ∠OmegaAB, ∠OmegaBC, and ∠OmegaCA are equal, with the unique such angle denoted omega. It is not a triangle center, but has trilinear coordinates


(Kimberling 1998, p. 47).

Note that extreme care is needed when consulting the literature, since reversing the order in which the points of the triangle are labeled results in exchanging the Brocard points.

The second Brocard point is the interior point Omega^' (also denoted tau_2 or Z_2) for which the angles ∠Omega^'AC, ∠Omega^'CB, and ∠Omega^'BA are equal, with the unique such angle denoted omega^'. It is not a triangle center, but has trilinear coordinates


(Kimberling 1998, p. 47).

Moreover, the two angles omega=omega^' are equal, and this angle is called the Brocard angle,


The first two Brocard points are isogonal conjugates (Johnson 1929, p. 266). They were described by French army officer Henri Brocard in 1875, although they had previously been investigated by Jacobi and, in 1816, Crelle (Wells 1991; Honsberger 1995, p. 98). They satisfy


where O is the circumcenter, R is the circumradius, and ∠OmegaOOmega^'=2omega, where O is the circumcenter and omega is the Brocard angle (Honsberger 1995, p. 106).

The common statement (Bernhart 1959; Wells 1991, pp. 21-22; Marshall et al. 2005) attributed to Brocard in response to an 1877 question from Edouard Lucas, namely that if three dogs start at the vertices of a triangle and chase either their left or right neighbor at a constant speed, the three will meet at either Omega or Omega^', is incorrect. This can be seen by considering an isosceles triangle that is nearly collinear and noting that two of the dogs will need to go much further than the other dog and so can't be traveling at the same speed (cf. Peterson 2001, Nester) in order to meet the other two at one of the points Omega or Omega^'.


One Brocard line, triangle median, and symmedian (out of the three of each) are concurrent, with AOmega, CK, and BG meeting at a point, where G is the triangle centroid and K is the symmedian point. Similarly, AOmega^', BG, and CK meet at a point which is the isogonal conjugate of the first (Johnson 1929, pp. 268-269; Honsberger 1995, pp. 121-124).


Let C_(BC) be the circle which passes through the vertices B and C and is tangent to the line AC at C, and similarly for C_(AB) and C_(BC). Then the circles C_(AB), C_(BC), and C_(AC) intersect in the first Brocard point Omega. Similarly, let C_(BC)^' be the circle which passes through the vertices B and C and is tangent to the line AB at B, and similarly for C_(AB)^' and C_(AC)^'. Then the circles C_(AB)^', C_(BC)^', and C_(AC)^' intersect in the second Brocard points Omega^' (Johnson 1929, pp. 264-265; Honsberger 1995, pp. 99-100).


The pedal triangles of Omega and Omega^' are congruent, and similar to the triangle DeltaABC (Johnson 1929, p. 269). Lengths involving the Brocard points include


Extend the segments AOmega, BOmega, and COmega to the circumcircle of DeltaABC to form DeltaC^'A^'B^', and the segments AOmega^', BOmega^', and COmega^' to form DeltaB^('')C^('')A^(''). Then DeltaA^'B^'C^' and DeltaA^('')B^('')C^('') are congruent to DeltaABC (Honsberger 1995, pp. 104-106).

The third Brocard point is related to a given triangle by the triangle center function


(Casey 1893, Kimberling 1994) and is Kimberling center X_(76). The third Brocard point Omega^('') (or tau_3 or Z_3) is collinear with the Spieker center and the isotomic conjugate of its triangle's incenter.

See also

Brocard Angle, Brocard Midpoint, Brocard Triangles, Equi-Brocard Center, First Brocard Point, Mice Problem, Pursuit Curve, Second Brocard Point, Third Brocard Point, Yff Points

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Bernhart, A. "Polygons of Pursuit." Scripta Math. 24, 23-50, 1959.Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 66, 1893.Coolidge, J. L. "The Brocard Figures." §1.5 in A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 60-84, 1971.Emmerich, A. Die Brocardschen Gebilde und ihre Beziehungen zu den verwandten merkwürdigen Punkten und Kreisen des Dreiecks. Berlin: Reimer, 1891.Gallatly, W. "The Brocard Points." §130 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 94, 1913.Honsberger, R. "The Brocard Points." Ch. 10 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 98-124, 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, 1929.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Encyclopedia of Triangle Centers: X(76)=3rd Brocard Point.", R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 65-66 and 79-80, 1893.Lemoine, É. "Propriétés relatives a deux points Omega, Omega^' du plan d'un triangle ABC qui se déduisent d'un point K quelconque di plan comme les points de Brocard de déduisent du point de Lemoine." Mathesis 6, Suppl. 3, 1-22, 1886.Marshall, J. A.; Broucke, M. E.; and Francis, B. A. "Pursuit Formations of Unicycles." Automata 41, 3-12, 2005., D. "Mathematics Seminar: Beetle Centers of Triangles." Software. "De punten van Brocard.", I. "MathTrek: Pursuing Pursuit Curves." Jul. 16, 2001., R. J. "Brocard Points, Circulant Matrices, and Descartes' Folium." Math. Mag. 61, 172-187, 1988.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 21-22, 1991.

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Brocard Points

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

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