“Brocard Angle”的概念、定义、习题解题方法及翻译-数学辞典

单词

Brocard Angle

释义

Brocard Angle

Define the first Brocard Point as the interior point of a Triangle for which theAngles , , and are equal. Similarly, define the secondBrocard Point as the interior point for which the Angles ,, and are equal. Then the Angles in both cases are equal, and this angle iscalled 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 sidelengths.

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.

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