## Brocard Angle

Define the first Brocard Point as the interior point of a Triangle for which the Angles , , and are equal. Similarly, define the second Brocard Point as the interior point for which the Angles , , and are equal. Then the Angles in both cases are equal, and this angle is called the Brocard angle, denoted .

The Brocard angle of a Triangle is given by the formulas

 (1) (2) (3) (4) (5) (6) (7)

where is the Triangle Area, , , and are Angles, and , , and are side lengths.

If an Angle of a Triangle is given, the maximum possible Brocard angle is given by

 (8)

Let a Triangle have Angles , , and . Then
 (9)

where
 (10)

(Le Lionnais 1983). This can be used to prove that
 (11)

(Abi-Khuzam 1974).

See also Brocard Circle, Brocard Line, Equi-Brocard Center, Fermat Point, Isogonic Centers

References

Abi-Khuzam, F. Proof of Yff's Conjecture on the Brocard Angle of a Triangle.'' Elem. Math. 29, 141-142, 1974.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 263-286 and 289-294, 1929.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 28, 1983.