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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

$\displaystyle \cot\omega$ $\textstyle =$ $\displaystyle \cot A+\cot B+\cot C$ (1)
  $\textstyle =$ $\displaystyle \left({a^2+b^2+c^2\over 4\Delta}\right)$ (2)
  $\textstyle =$ $\displaystyle {1+\cos\alpha_1\cos\alpha_2\cos\alpha_3\over \sin\alpha_1\sin\alpha_2\sin\alpha_3}$ (3)
  $\textstyle =$ $\displaystyle {\sin^2\alpha_1+\sin^2\alpha_2+\sin^2\alpha_3\over 2\sin\alpha_1\sin\alpha_2\sin\alpha_3}$ (4)
  $\textstyle =$ $\displaystyle {a_1\sin\alpha_1+a_2\sin\alpha_2+a_3\sin\alpha_3\over a_1\cos\alpha_1+a_2\cos\alpha_2+a_3\cos\alpha_3}$ (5)
$\displaystyle \csc^2\omega$ $\textstyle =$ $\displaystyle \csc^2\alpha_1+\csc^2\alpha_2+\csc^2\alpha_3$ (6)
$\displaystyle \sin\omega$ $\textstyle =$ $\displaystyle {2\Delta \over\sqrt{{a_1}^2{a_2}^2+{a_2}^2{a_3}^2+{a_3}^2{a_1}^2}},$ (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).
\end{displaymath} (8)

Let a Triangle have Angles $A$, $B$, and $C$. Then
\sin A\sin B\sin C\leq kABC,
\end{displaymath} (9)

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

(Le Lionnais 1983). This can be used to prove that
\end{displaymath} (11)

(Abi-Khuzam 1974).

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


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.

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