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
Similarly,
From this it follows that
\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
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,
Comparing with the above equality, we obtain
Adding 1 to both sides and using <math>AF'+F'B=AF+FB=AB<math>, we obtain
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
- Ceva's Theorem, Interactive proof with animation and key concepts (http://agutie.homestead.com/files/ceva.htm) by Antonio Gutierrez from the land of the Incas
- Derivations and applications of Ceva's Theorem (http://www.cut-the-knot.org/Generalization/ceva.shtml)
- Cevian Nest (http://www.cut-the-knot.org/Curriculum/Geometry/CevaNest.shtml)
- Trigonometric Form of Ceva's Theorem (http://www.cut-the-knot.org/triangle/TrigCeva.shtml)de:Satz von Ceva
ja:チェバの定理 pl:Twierdzenie Cevy ru:Теорема Чевы sl:Cevov izrek zh:塞瓦定理 fi:Cevan lause