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Brocard Circle

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The Circle passing through the first and second Brocard Points $\Omega$ and $\Omega'$, the Lemoine Point $K$, and the Circumcenter $O$ of a given Triangle. The Brocard Points $\Omega$ and $\Omega'$ are symmetrical about the Line ${\leftrightarrow\atop{\displaystyle KO}}$, which is called the Brocard Line. The Line Segment $\overline{KO}$ is called the Brocard Diameter, and it has length


where $R$ is the Circumradius and $\omega$ is the Brocard Angle. The distance between either of the Brocard Points and the Lemoine Point is

\overline{\Omega K}=\overline{\Omega'K}=\overline{\Omega O}\tan\omega.

See also Brocard Angle, Brocard Diameter, Brocard Points


Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 272, 1929.

© 1996-9 Eric W. Weisstein