## Brocard Points The first Brocard point is the interior point (or or ) of a Triangle for which the Angles , , and are equal. The second Brocard point is the interior point (or or ) for which the Angles , , and are equal. The Angles in both cases are equal to the Brocard Angle ,     The first two Brocard points are Isogonal Conjugates (Johnson 1929, p. 266). Let be the Circle which passes through the vertices and and is Tangent to the line at , and similarly for and . Then the Circles , , and intersect in the first Brocard point . Similarly, let be the Circle which passes through the vertices and and is Tangent to the line at , and similarly for and . Then the Circles , , and intersect in the second Brocard points (Johnson 1929, pp. 264-265). The Pedal Triangles of and are congruent, and Similar to the Triangle (Johnson 1929, p. 269). Lengths involving the Brocard points include (1) (2)

Brocard's third point is related to a given Triangle by the Triangle Center Function (3)

(Casey 1893, Kimberling 1994). The third Brocard point (or or ) is Collinear with the Spieker Center and the Isotomic Conjugate Point of its Triangle's Incenter.

References

Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 66, 1893.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 263-286, 1929.

Kimberling, C. Central Points and Central Lines in the Plane of a Triangle.'' Math. Mag. 67, 163-187, 1994.

Stroeker, R. J. Brocard Points, Circulant Matrices, and Descartes' Folium.'' Math. Mag. 61, 172-187, 1988.