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Ceva's theorem

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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

:$frac\{AF\}\{FB\}\; cdot\; frac\{BD\}\{DC\}\; cdot\; frac\{CE\}\{EA\}\; =\; 1,$ where AF indicates the directed distance between A and F (i.e. distance in one direction along a line is counted as positive, and in the other direction is counted as negative).

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

:$frac\{sinangle\; BAD\}\{sinangle\; CAD\}timesfrac\{sinangle\; ACF\}\{sinangle\; BCF\}timesfrac\{sinangle\; CBE\}\{sinangle\; ABE\}=1.$

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.)

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{|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 $F\; \text{'}$. By the direction we have just proven,

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

Comparing with the above equality, we obtain

: $frac\{AF\; \text{'}\}\{F\; \text{'}B\}=frac\{AF\}\{FB\}.$

Adding 1 to both sides and using $AF\; \text{'}$F 'B=AF FB = AB (case 1), or subtracting both sides from 1 and using $F\; \text{'}B\; -\; AF\; \text{'}\; =\; FB\; -\; AF\; =\; AB$ (case 2) we obtain

: $frac\{AB\}\{F\; \text{'}B\}=frac\{AB\}\{FB\}.$

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

For the trigonometric 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: :$frac\{sinangle\; OAB\}\{sinangle\; OBA\}=frac\{OB\}\{OA\}\; text\{\; ;\; \}\; frac\{sinangle\; OBC\}\{sinangle\; OCB\}=frac\{OC\}\{OB\}text\{\; ;\; \}\; frac\{sinangle\; OCA\}\{sinangle\; OAC\}=frac\{OA\}\{OC\}.$

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

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.)

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 . The theorem has also been generalized to triangles on other surfaces of constant curvature (Masal'tsev 1994).

See also

• Menelaus' theorem - the dual of Ceva's theorem
• Projective geometry
• Median (geometry) - an application

References

• .
• J. B. Hogendijk, "Al-Mutaman ibn Hűd, 11the century king 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

External links

• »Menelaus and Ceva at MathPages
• »Derivations and applications of Ceva's Theorem at cut-the-knot
• »Trigonometric Form of Ceva's Theorem at cut-the-knot
• »Glossary of Encyclopedia of Triangle Centers includes definitions of cevian triangle, cevian nest, anticevian triangle, Ceva conjugate, and cevapoint
• »Conics Associated with a Cevian Nest, by Clark Kimberling
• '' »Ceva's Theorem'' by Jay Warendorff, Wolfram Demonstrations Project.

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