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Brocard Circle


BrocardCircle

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

 l=-(bc)/(a^2+b^2+c^2),
(1)

corresponding to the triangle centroid G and giving trilinear equation

 abc(alpha^2+beta^2+gamma^2)=a^3betagamma+b^3gammaalpha+c^3alphabeta
(2)

(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

R_B=sqrt((a^4+b^4+c^4)-(a^2b^2+b^2c^2+c^2a^2))R
(3)
=(Rsqrt(1-4sin^2omega))/(2cosomega),
(4)

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

 OmegaK^_=Omega^'K^_=OmegaO^_tanomega.
(5)

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

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References

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.

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Brocard Circle

Cite this as:

Weisstein, Eric W. "Brocard Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BrocardCircle.html

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