Ceva's theorem
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Ceva's theorem is a very popular 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 \cdot \frac \cdot \frac = 1.]
[\frac\times\frac\times\frac=1].
It was first proven by Giovanni Ceva in his 1678 work De lineis rectis.
Proof
Suppose [AD], [BE] and [CF] intersect at a point [O]. Because [\triangle BOD] and [\triangle COD] have the same height, we have
Similarly,
From this it follows that
Similarly,
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.
See also
External links
- [Ceva's Theorem, Interactive proof with animation and key concepts] by Antonio Gutierrez from the land of the Incas
- [Derivations and applications of Ceva's Theorem] at cut-the-knot
- [Cevian Nest] at cut-the-knot
- [Trigonometric Form of Ceva's Theorem] at cut-the-knot
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