Ceva's theorem is a 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
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
The theorem was proved by Giovanni Ceva in his 1678 work De lineis rectis, but it was also proven much earlier by Yusuf AlMu'taman ibn Hűd, an eleventhcentury 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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Proof of the theorem
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 (case 1), or subtracting both sides from 1 and using F'B − AF' = FB − AF = AB (case 2) 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.
For the trigonometric form of the theorem, one approach is to view the three cevians, concurrent at point O, as partitioning the triangle into three smaller triangles: ,, and .
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