The Brocard circle, also known as the sevenpoint circle, is the circle having the line segment connecting the circumcenter and symmedian
point
of a triangle as its diameter (known as the Brocard
diameter). This circle also passes through the first
and second Brocard points and , respectively. It also passes through Kimberling
centers
for ,
6, 1083, and 1316.
It has circle function

(1)

corresponding to the triangle centroid and giving trilinear equation

(2)

(Carr 1970; Kimberling 1998, p. 233).
The Brocard points and are symmetrical about the line , which is called the Brocard line. The line segment is called the Brocard
diameter, which has length twice the Brocard circle radius , where
with
the circumradius and the Brocard angle of
the reference triangle.
The center of the Brocard circle is the Brocard midpoint .
The distance between either of the Brocard points
and the symmedian point is

(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 SciencesCongrés
d'Alger 10, 138159, 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. 101102,
1913.Honsberger, R. "The Brocard Circle." §10.3 in Episodes
in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math.
Assoc. Amer., pp. 106110, 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, 1295, 1998.Lachlan,
R. "The Brocard Circle." §134135 in An
Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 7881,
1893.Referenced on WolframAlpha
Brocard Circle
Cite this as:
Weisstein, Eric W. "Brocard Circle." From
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