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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

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

.

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

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.)

Contents 1 Proof of the theorem 2 See also 3 External links 4 References

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

Similarly,

From this it follows that

Similarly,

and

Multiplying these three equations gives

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,

Comparing with the above equality, we obtain

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

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.

• Menelaus' theorem

• Ceva's Theorem, Interactive proof with animation and key concepts by Antonio Gutierrez, Peru.
• 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

• J. B. Hogendijk, "Al-Mutaman ibn Hűd, 11the century kin of Saragossa and brilliant mathematician," Historia Mathematica 22 (1995) 1-18.