Ceva's theorem

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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 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


From this it follows that



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'BAF' = FBAF = 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 \triangle ABC into three smaller triangles: \triangle AOB,\triangle BOC, and \triangle COA.

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