Ceva's theorem

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Ceva's Theorem (pronounced "Cheva") 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

<math>\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1.<math>

It was first proved by Giovanni Ceva.

Missing image
Cevastheorem.jpg


Proof

Suppose <math>AD<math>, <math>BE<math> and <math>CF<math> intersect at a point <math>X<math>. Because <math>\triangle BXD<math> and <math>\triangle CXD<math> have the same height, we have

<math>\frac{|\triangle BXD|}{|\triangle CXD|}=\frac{BD}{DC}.<math>

Similarly,

<math>\frac{|\triangle BAD|}{|\triangle CAD|}=\frac{BD}{DC}.<math>

From this it follows that

<math>\frac{BD}{DC}=

\frac{|\triangle BAD|-|\triangle BXD|}{|\triangle CAD|-|\triangle CXD|}

=\frac{|\triangle ABX|}{|\triangle CAX|}.<math>

Similarly,

<math>\frac{CE}{EA}=\frac{|\triangle BCX|}{|\triangle ABX|}<math>, and
<math>\frac{AF}{FB}=\frac{|\triangle CAX|}{|\triangle BCX|}<math>.

Multiplying these three equations gives

<math>\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1<math>

as required. Conversely, suppose that the points <math>D<math>, <math>E<math> and <math>F<math> satisfy the above equality. Let <math>AD<math> and <math>BE<math> intersect at <math>X<math>, and let <math>CX<math> intersect <math>AB<math> at <math>F'<math>. By the direction we have just proven,

<math>\frac{AF'}{F'B} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1.<math>

Comparing with the above equality, we obtain

<math>\frac{AF'}{F'B}=\frac{AF}{FB}.<math>

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

<math>\frac{AB}{F'B}=\frac{AB}{FB}.<math>

Thus <math>F'B=FB<math>, so that <math>F<math> and <math>F'<math> coincide (recalling that the distances are directed). Therefore <math>AD<math>, <math>BE<math> and <math>CF<math>=<math>CF'<math> intersect at <math>X<math>, and both implications are proven.

See also

External links

ja:チェバの定理 pl:Twierdzenie Cevy ru:Теорема Чевы sl:Cevov izrek zh:塞瓦定理 fi:Cevan lause

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