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 Math Help > Geometry > Polygons and Triangles > Ceva's Theorem

Ceva's Theorem

This theorem was proved by Giovanni Ceva (1648-1734).

 Ceva's theorem states that given three arbitrary cevians AD, BE and CF, the three of them all meet at a point P if and only if (1)  AF/FB � BD/DC � CE/EA = 1 (The lines that meet at a point are said to be concurrent)

Proof:

 Extend the lines BE and CF beyond the triangle until they meet GH, the line through A parallel to BC. There are several pairs of similar triangles: AHF and BCF, AGE and CBE, AGP and DBP, DCP and AHP. From these and in that order we derive the following proportions: AF/FB=AH/BC (*)  CE/EA=BC/AG (*)  AG/BD=AP/DP  AH/DC=AP/DP  from the last two we conclude that AG/BD = AH/DC and, hence,  BD/DC = AG/AH (*).

Multiplying the identities marked with (*) we get

AF/FB � BD/DC � CE/EA = AH/BC � BC/AG � AG/AH = (AH�BC�AG)/(BC�AG�AH) = 1

Therefore, if the lines AD, BE and CF intersect at a single point P, the identity (1) does hold. Which is to say that the fact of the three lines intersecting at one point is sufficient for the condition (1) to hold. Let us now prove that it's also necessary. This would constitute the second part of the theorem. In other words, let us prove that if (1) holds then AD, BE, CF are concurrent.

Indeed, assume that P is the point of intersection of BE and CF and draw the line AP until its intersection with BC at a point D'. Then, from the just proven part of the theorem it follows that

AF/FB � BD'/D'C � CE/EA = 1

On the other hand, it's given that

AF/FB � BD/DC � CE/EA = 1

Combining the two we get

BD'/D'C = BD/DC or
BD'/D'C + 1 = BD/DC + 1 or
(BD'+D'C)/D'C = (BD+DC)/DC

Finally

BC/D'C = BC/DC

which immediately implies D'C=DC. That is, D' and D are one and the same point.

Q.E.D.

"Area" and "Angle" Forms of Ceva's Theorem

 By Ceva's theorem, the three cevians AD, BE and CF all meet at a point P iff (1)  AF/FB � BD/DC � CE/EA = 1. The "area" form of Ceva's theorem is an immediate corollary, stating that three cevians meet at a point iff the product of the ratios of the areas (2) (area ACF) / (area FCB) � (area BAD) / (area DAC) � (area CBE) / (area EBA) = 1, which follows from (1) because each pair of triangles have the same height. Using the sine formula for the area of a triangle, equation 2 becomes (3) (AC CF sin ACF) / (FC CB sin FCB) � (BA AD sin BAD) / (DA AC sin DAC) � (CB BE sin CBE) / (EB BA sin EBA) = 1 The lengths of the sides of triangle ABC all cancel, leaving us with (4) (sin ACF) / (sin FCB) � (sin BAD) / (sin DAC) � (sin CBE) / (sin EBA) = 1

Internet references

http://www.cut-the-knot.com/Generalization/ceva.html has a java-enabled diagram on which you can drag the vertices of the triangle, and watch the various similar triangles in action.

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Summary of geometrical theorems