The Brocard circle, also known as the seven-point 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 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. https://mathworld.wolfram.com/BrocardCircle.html
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