“Ceva's Theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Ceva's Theorem

释义

Ceva's Theorem

Given a Triangle with Vertices , , and and points along the sides , , and, a Necessary and Sufficient condition for the Cevians , , and to beConcurrent (intersect in a single point) is that

(1)

Let

be an arbitrary

-gon,

a given point, and

a Positive Integer such that

. For

, ...,

, let

be the intersection of the lines

and

, then

(2)

Here,

and

(3)

is the Ratio of the lengths

and

with a plus or minus sign depending on whether these segments havethe same or opposite directions (Grünbaum and Shepard 1995).

Another form of the theorem is that three Concurrent lines from the Vertices of aTriangle divide the opposite sides in such fashion that the product of three nonadjacent segments equals the productof the other three (Johnson 1929, p. 147).

See also Hoehn's Theorem, Menelaus' Theorem

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 122, 1987.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 4-5, 1967.

Grünbaum, B. and Shepard, G. C. ``Ceva, Menelaus, and the Area Principle.'' Math. Mag. 68, 254-268, 1995.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 145-151, 1929.

Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., p. xx, 1995.

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