Ceva's theorem

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For other uses, see Ceva (disambiguation).
Ceva's theorem, case 1: the three lines are concurrent at a point O inside ABC
Ceva's theorem, case 1: the three lines are concurrent at a point O inside ABC
Ceva's theorem, case 2: the three lines are concurrent at a point O outside ABC
Ceva's theorem, case 2: the three lines are concurrent at a point O outside ABC

Ceva's theorem is a well-known theorem in elementary geometry. Given a triangle ABC, and points D, E, and F that lie on lines BC, CA, and AB respectively, the theorem states that lines AD, BE and CF are concurrent if and only if

\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1.

There is also an equivalent trigonometric form of Ceva's Theorem, that is, AD,BE,CF concur if and only if

\frac{\sin\angle BAD}{\sin\angle CAD}\times\frac{\sin\angle ACF}{\sin\angle BCF}\times\frac{\sin\angle CBE}{\sin\angle ABE}=1.

The theorem was proved by Giovanni Ceva in his 1678 work De lineis rectis, but it was also proved much earlier by Yusuf Al-Mu'taman ibn Hűd, an eleventh-century king of Zaragoza.

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

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[edit] Proof of the theorem

Suppose AD, BE and CF intersect at a point O. Because \triangle BOD and \triangle COD have the same height, we have

\frac{|\triangle BOD|}{|\triangle COD|}=\frac{BD}{DC}.

Similarly,

\frac{|\triangle BAD|}{|\triangle CAD|}=\frac{BD}{DC}.

From this it follows that

\frac{BD}{DC}= \frac{|\triangle BAD|-|\triangle BOD|}{|\triangle CAD|-|\triangle COD|} =\frac{|\triangle ABO|}{|\triangle CAO|}.

Similarly,

\frac{CE}{EA}=\frac{|\triangle BCO|}{|\triangle ABO|},

and

\frac{AF}{FB}=\frac{|\triangle CAO|}{|\triangle BCO|}.

Multiplying these three equations gives

\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1,

as required. Conversely, suppose that the points D, E and F satisfy the above equality. Let AD and BE intersect at O, and let CO intersect AB at F'. By the direction we have just proven,

\frac{AF '}{F 'B} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1.

Comparing with the above equality, we obtain

\frac{AF '}{F 'B}=\frac{AF}{FB}.

Adding 1 to both sides and using AF' + F'B = AF + FB = AB, we obtain

\frac{AB}{F 'B}=\frac{AB}{FB}.

Thus F'B = FB, so that F and F' coincide (recalling that the distances are directed). Therefore AD, BE and CF = CF' intersect at O, and both implications are proven.

For the trigonometic form of the theorem, one approach is to view the three cevians, concurrent at point O, as partitioning the triangle \triangle ABC into three smaller triangles: \triangle AOB,\triangle BOC, and \triangle COA.

Applying the law of sines to each triangle we get:

\frac{\sin\angle OAB}{\sin\angle OBA}=\frac{OB}{OA} \text{  ; } \frac{\sin\angle OBC}{\sin\angle OCB}=\frac{OC}{OB}\text{  ; } \frac{\sin\angle OCA}{\sin\angle OAC}=\frac{OA}{OC}.

When we mutiply the three equations, the right side will equal 1. The six sines on the left side, when rearranged, will yield the expression given in the theorem.

[edit] Generalizations

The theorem can be generalized to higher dimensional simplexes using barycentric coordinates. Define a cevian of an n-simplex as a ray from each vertex to a point on the opposite (n-1)-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. (Landy. See Wernicke for an earlier result.)

The analogue of the theorem for general polygons in the plane has been known since the early nineteenth century (Grünbaum & Shephard 1995, p. 266). The theorem has also been generalized to triangles on other surfaces of constant curvature (Masal'tsev 1994).

[edit] See also

[edit] References

  • Grünbaum, Branko & Shephard, G. C. (1995), “Ceva, Menelaus and the Area Principle”, Mathematics Magazine 68 (4): 254–268, <http://links.jstor.org/sici?sici=0025-570X(199510)68%3A4%3C254%3ACMATAP%3E2.0.CO%3B2-0> .
  • J. B. Hogendijk, "Al-Mutaman ibn Hűd, 11the century kin of Saragossa and brilliant mathematician," Historia Mathematica 22 (1995) 1-18.
  • Landy, Steven. A Generalization of Ceva's Theorem to Higher Dimensions. The American Mathematical Monthly, Vol. 95, No. 10 (Dec., 1988), pp. 936-939
  • Masal'tsev, L. A. (1994) "Incidence theorems in spaces of constant curvature." Journal of Mathematical Sciences, Vol. 72, No. 4
  • Wernicke, Paul. The Theorems of Ceva and Menelaus and Their Extension. The American Mathematical Monthly, Vol. 34, No. 9 (Nov., 1927), pp. 468-472

[edit] External links