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) |

**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.

© 1996-9

1999-05-26