“Brocard Points”的概念、定义、习题解题方法及翻译-数学辞典

单词

Brocard Points

释义

Brocard Points

The first Brocard point is the interior point (or or ) of a Triangle for which theAngles , , and are equal. The second Brocard point isthe interior point (or or ) for which the Angles ,, and are equal. The Angles in both cases are equal to theBrocard 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 thefirst Brocard point . Similarly, let be the Circle which passes through the vertices and andis 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 theTriangle (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)

(Casey 1893, Kimberling 1994).The third Brocard point

(or

) is Collinear with the Spieker Center and theIsotomic Conjugate Point of its Triangle's Incenter.

See also Brocard Angle, Brocard Midpoint, Equi-Brocard Center, Yff Points

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.

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