Brocard Angle

Define the first Brocard Point as the interior point $\Omega$ of a Triangle for which the Angles $\angle\Omega AB$, $\angle\Omega BC$, and $\angle\Omega CA$ are equal. Similarly, define the second Brocard Point as the interior point $\Omega'$ for which the Angles $\angle\Omega'AC$, $\angle\Omega'CB$, and $\angle\Omega'BA$ are equal. Then the Angles in both cases are equal, and this angle is called the Brocard angle, denoted $\omega$.

The Brocard angle $\omega$ of a Triangle $\Delta ABC$ is given by the formulas

$$\cot\omega = \cot A+\cot B+\cot C$$ (1)

$$= \left({a^2+b^2+c^2\over 4\Delta}\right)$$ (2)

$$= {1+\cos\alpha_1\cos\alpha_2\cos\alpha_3\over \sin\alpha_1\sin\alpha_2\sin\alpha_3}$$ (3)

$$= {\sin^2\alpha_1+\sin^2\alpha_2+\sin^2\alpha_3\over 2\sin\alpha_1\sin\alpha_2\sin\alpha_3}$$ (4)

$$= {a_1\sin\alpha_1+a_2\sin\alpha_2+a_3\sin\alpha_3\over a_1\cos\alpha_1+a_2\cos\alpha_2+a_3\cos\alpha_3}$$ (5)

$$\csc^2\omega = \csc^2\alpha_1+\csc^2\alpha_2+\csc^2\alpha_3$$ (6)

$$\sin\omega = {2\Delta \over\sqrt{{a_1}^2{a_2}^2+{a_2}^2{a_3}^2+{a_3}^2{a_1}^2}},$$ (7)

where $\Delta$ is the Triangle Area, $A$, $B$, and $C$ are Angles, and $a$, $b$, and $c$ are side lengths.

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

$$\cot\omega={\textstyle{3\over 2}}\tan({\textstyle{1\over 2}}\alpha)+{\textstyle{1\over 2}}\cos({\textstyle{1\over 2}}\alpha).$$ (8)

Let a Triangle have Angles $A$, $B$, and $C$. Then

$$\sin A\sin B\sin C\leq kABC,$$ (9)

where

$$k=\left({3\sqrt{3}\over 2\pi}\right)^3$$ (10)

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

$$8\omega^3<ABC$$ (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.