Brocard Circle


The Brocard circle, also known as the seven-point circle, is the circle having the line segment connecting the circumcenter O and symmedian point K of a triangle DeltaABC as its diameter (known as the Brocard diameter). This circle also passes through the first and second Brocard points Omega and Omega^', respectively. It also passes through Kimberling centers X_i for i=3, 6, 1083, and 1316.

It has circle function


corresponding to the triangle centroid G and giving trilinear equation


(Carr 1970; Kimberling 1998, p. 233).

The Brocard points Omega and Omega^' are symmetrical about the line <->; KO, which is called the Brocard line. The line segment KO^_ is called the Brocard diameter, which has length twice the Brocard circle radius R_B, where


with R the circumradius and omega the Brocard angle of the reference triangle.

The center of the Brocard circle is the Brocard midpoint X_(182).

The distance between either of the Brocard points and the symmedian point is


The Brocard circle and first Lemoine circle are concentric.

It is orthogonal to the Parry circle.

See also

Brocard Angle, Brocard Diameter, Brocard Line, Brocard Points, Brocard Triangles, Cosine Circle

Explore with Wolfram|Alpha


Brocard, M. H. "Etude d'un nouveau cercle du plan du triangle." Assoc. Français pour l'Academie des Sciences-Congrés d'Alger 10, 138-159, 1881.Carr, G. S. Art. 4754c in Synopsis of Elementary Results in Pure Mathematics, 2nd ed., 2 vols. New York: Chelsea, 1970.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 75, 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, pp. 101-102, 1913.Honsberger, R. "The Brocard Circle." §10.3 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 106-110, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 272, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Lachlan, R. "The Brocard Circle." §134-135 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 78-81, 1893.

Referenced on Wolfram|Alpha

Brocard Circle

Cite this as:

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

Subject classifications