Английская Википедия:Ceva's theorem

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Файл:Ceva's theorem 1.svg
Ceva's theorem, case 1: the three lines are concurrent at a point Шаблон:Mvar inside Шаблон:Math
Файл:Ceva's theorem 2.svg
Ceva's theorem, case 2: the three lines are concurrent at a point Шаблон:Mvar outside Шаблон:Math

In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle Шаблон:Math, let the lines Шаблон:Mvar be drawn from the vertices to a common point Шаблон:Mvar (not on one of the sides of Шаблон:Math), to meet opposite sides at Шаблон:Mvar respectively. (The segments Шаблон:Mvar are known as cevians.) Then, using signed lengths of segments,

<math>\frac{\overline{AF}}{\overline{FB}} \cdot \frac{\overline{BD}}{\overline{DC}} \cdot \frac{\overline{CE}}{\overline{EA}} = 1.</math>

In other words, the length Шаблон:Mvar is taken to be positive or negative according to whether Шаблон:Mvar is to the left or right of Шаблон:Mvar in some fixed orientation of the line. For example, Шаблон:Mvar is defined as having positive value when Шаблон:Mvar is between Шаблон:Mvar and Шаблон:Mvar and negative otherwise.

Ceva's theorem is a theorem of affine geometry, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two line segments that are collinear). It is therefore true for triangles in any affine plane over any field.

A slightly adapted converse is also true: If points Шаблон:Mvar are chosen on Шаблон:Mvar respectively so that

<math>\frac{\overline{AF}}{\overline{FB}} \cdot \frac{\overline{BD}}{\overline{DC}} \cdot \frac{\overline{CE}}{\overline{EA}} = 1,</math>

then Шаблон:Mvar are concurrent, or all three parallel. The converse is often included as part of the theorem.

The theorem is often attributed to Giovanni Ceva, who published it in his 1678 work De lineis rectis. But it was proven much earlier by Yusuf Al-Mu'taman ibn Hűd, an eleventh-century king of Zaragoza.[1]

Associated with the figures are several terms derived from Ceva's name: cevian (the lines Шаблон:Mvar are the cevians of Шаблон:Mvar), cevian triangle (the triangle Шаблон:Math is the cevian triangle of Шаблон:Mvar); cevian nest, anticevian triangle, Ceva conjugate. (Ceva is pronounced Chay'va; cevian is pronounced chev'ian.)

The theorem is very similar to Menelaus' theorem in that their equations differ only in sign. By re-writing each in terms of cross-ratios, the two theorems may be seen as projective duals.[2]

Proofs

Several proofs of the theorem have been given.[3][4] Two proofs are given in the following.

The first one is very elementary, using only basic properties of triangle areas.[3] However, several cases have to be considered, depending on the position of the point Шаблон:Mvar.

The second proof uses barycentric coordinates and vectors, but is somehow more natural and not case dependent. Moreover, it works in any affine plane over any field.

Using triangle areas

First, the sign of the left-hand side is positive since either all three of the ratios are positive, the case where Шаблон:Mvar is inside the triangle (upper diagram), or one is positive and the other two are negative, the case Шаблон:Mvar is outside the triangle (lower diagram shows one case).

To check the magnitude, note that the area of a triangle of a given height is proportional to its base. So

<math>\frac{|\triangle BOD|}{|\triangle COD|}=\frac{\overline{BD}}{\overline{DC}}=\frac{|\triangle BAD|}{|\triangle CAD|}.</math>

Therefore,

<math>\frac{\overline{BD}}{\overline{DC}}=

\frac{|\triangle BAD|-|\triangle BOD|}{|\triangle CAD|-|\triangle COD|} =\frac{|\triangle ABO|}{|\triangle CAO|}.</math> (Replace the minus with a plus if Шаблон:Mvar and Шаблон:Mvar are on opposite sides of Шаблон:Mvar.) Similarly,

<math>\frac{\overline{CE}}{\overline{EA}}=\frac{|\triangle BCO|}{|\triangle ABO|},</math>

and

<math>\frac{\overline{AF}}{\overline{FB}}=\frac{|\triangle CAO|}{|\triangle BCO|}.</math>

Multiplying these three equations gives

<math>\left|\frac{\overline{AF}}{\overline{FB}} \cdot \frac{\overline{BD}}{\overline{DC}} \cdot \frac{\overline{CE}}{\overline{EA}} \right|= 1,</math>

as required.

The theorem can also be proven easily using Menelaus' theorem.[5] From the transversal Шаблон:Mvar of triangle Шаблон:Math,

<math>\frac{\overline{AB}}{\overline{BF}} \cdot \frac{\overline{FO}}{\overline{OC}} \cdot \frac{\overline{CE}}{\overline{EA}} = -1</math>

and from the transversal Шаблон:Math of triangle Шаблон:Math,

<math>\frac{\overline{BA}}{\overline{AF}} \cdot \frac{\overline{FO}}{\overline{OC}} \cdot \frac{\overline{CD}}{\overline{DB}} = -1.</math>

The theorem follows by dividing these two equations.

The converse follows as a corollary.[3] Let Шаблон:Mvar be given on the lines Шаблон:Mvar so that the equation holds. Let Шаблон:Mvar meet at Шаблон:Mvar and let Шаблон:Mvar be the point where Шаблон:Mvar crosses Шаблон:Mvar. Then by the theorem, the equation also holds for Шаблон:Mvar. Comparing the two,

<math>\frac{\overline{AF}}{\overline{FB}} = \frac{\overline{AF'}}{\overline{F'B}}</math>

But at most one point can cut a segment in a given ratio so Шаблон:Mvar.

Using barycentric coordinates

Given three points Шаблон:Mvar that are not collinear, and a point Шаблон:Mvar, that belongs to the same plane, the barycentric coordinates of Шаблон:Mvar with respect of Шаблон:Mvar are the unique three numbers <math>\lambda_A, \lambda_B, \lambda_C</math> such that

<math>\lambda_A + \lambda_B + \lambda_C =1,</math>

and

<math>\overrightarrow{XO}=\lambda_A\overrightarrow{XA} + \lambda_B\overrightarrow{XB} + \lambda_C\overrightarrow{XC},</math>

for every point Шаблон:Mvar (for the definition of this arrow notation and further details, see Affine space).

For Ceva's theorem, the point Шаблон:Mvar is supposed to not belong to any line passing through two vertices of the triangle. This implies that <math>\lambda_A \lambda_B \lambda_C\ne 0.</math>

If one takes for Шаблон:Mvar the intersection Шаблон:Mvar of the lines Шаблон:Mvar and Шаблон:Mvar (see figures), the last equation may be rearranged into

<math>\overrightarrow{FO}-\lambda_C\overrightarrow{FC}=\lambda_A\overrightarrow{FA} + \lambda_B\overrightarrow{FB}.</math>

The left-hand side of this equation is a vector that has the same direction as the line Шаблон:Mvar, and the right-hand side has the same direction as the line Шаблон:Mvar. These lines have different directions since Шаблон:Mvar are not collinear. It follows that the two members of the equation equal the zero vector, and

<math>\lambda_A\overrightarrow{FA} + \lambda_B\overrightarrow{FB}=0.</math>

It follows that

<math>\frac{\overline{AF}}{\overline{FB}}=\frac{\lambda_B}{\lambda_A},</math>

where the left-hand-side fraction is the signed ratio of the lengths of the collinear line segments Шаблон:Mvar and Шаблон:Mvar.

The same reasoning shows

<math>\frac{\overline{BD}}{\overline{DC}}=\frac{\lambda_C}{\lambda_B}\quad \text{and}\quad \frac{\overline{CE}}{\overline{EA}}=\frac{\lambda_A}{\lambda_C}.</math>

Ceva's theorem results immediately by taking the product of the three last equations.

Generalizations

The theorem can be generalized to higher-dimensional simplexes using barycentric coordinates. Define a cevian of an Шаблон:Mvar-simplex as a ray from each vertex to a point on the opposite (Шаблон:Math)-face (facet). Then the cevians are concurrent if and only if a mass distribution can be assigned to the vertices such that each cevian intersects the opposite facet at its center of mass. Moreover, the intersection point of the cevians is the center of mass of the simplex.[6][7]

Another generalization to higher-dimensional simplexes extends the conclusion of Ceva's theorem that the product of certain ratios is 1. Starting from a point in a simplex, a point is defined inductively on each Шаблон:Mvar-face. This point is the foot of a cevian that goes from the vertex opposite the Шаблон:Mvar-face, in a (Шаблон:Math)-face that contains it, through the point already defined on this (Шаблон:Math)-face. Each of these points divides the face on which it lies into lobes. Given a cycle of pairs of lobes, the product of the ratios of the volumes of the lobes in each pair is 1.[8]

Routh's theorem gives the area of the triangle formed by three cevians in the case that they are not concurrent. Ceva's theorem can be obtained from it by setting the area equal to zero and solving.

The analogue of the theorem for general polygons in the plane has been known since the early nineteenth century.[9] The theorem has also been generalized to triangles on other surfaces of constant curvature.[10]

The theorem also has a well-known generalization to spherical and hyperbolic geometry, replacing the lengths in the ratios with their sines and hyperbolic sines, respectively.

See also

References

Шаблон:Reflist

Further reading

External links

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  1. ↑ Шаблон:Cite book
  2. ↑ Шаблон:Cite journal
  3. ↑ 3,0 3,1 3,2 Шаблон:Cite book
  4. ↑ Alfred S. Posamentier and Charles T. Salkind (1996), Challenging Problems in Geometry, pages 177–180, Dover Publishing Co., second revised edition.
  5. ↑ Follows Шаблон:Cite book
  6. ↑ Шаблон:Cite journal
  7. ↑ Шаблон:Cite journal
  8. ↑ Шаблон:Cite journal
  9. ↑ Шаблон:Cite journal
  10. ↑ Шаблон:Cite journal
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