Theorem of Ceva

Theorem of Ceva
${\displaystyle O=AD\cap CF\cap BE\,\Rightarrow \,{\frac {|AE|\cdot |BD|\cdot |CF|}{|BE|\cdot |CD|\cdot |AF|}}=1}$

Ceva's theorem with parallel lines
${\displaystyle O=AD\parallel CF\parallel BE\,\Rightarrow \,{\frac {|AE|\cdot |BD|\cdot |CF|}{|BE|\cdot |CD|\cdot |AF|}}=1}$

The set of Ceva is a geometric statement about Ecktransversalen in the triangle that the Italian mathematician Giovanni Ceva (1647-1734) in 1678 in his work De lineis rectis proved. The sentence was already described in the 11th century by the mathematician and emir of Zaragossa Yusuf al-Mutaman .

In a triangle are , and three corner transversals ( i.e. connecting lines between a corner and a point on the opposite side or their extension) that intersect at a point inside or outside the triangle. Then: ${\displaystyle ABC}$ ${\displaystyle AD}$ ${\displaystyle BE}$ ${\displaystyle CF}$ ${\displaystyle O}$

${\displaystyle TV(A,B,F)\cdot TV(B,C,D)\cdot TV(C,A,E)=1}$

Here is the (oriented, ie possibly negative) division ratio of which three on a straight points lying with is defined by . If it is between and , the mentioned division ratio is the same , otherwise the same . ${\displaystyle TV(U,V,W)}$ ${\displaystyle U,V,W}$ ${\displaystyle U,V,W}$ ${\displaystyle W\neq V}$ ${\displaystyle {\overrightarrow {UW}}=TV(U,V,W)\cdot {\overrightarrow {WV}}}$ ${\displaystyle W}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle {\overline {UW}}/{\overline {WV}}}$ ${\displaystyle -{\overline {UW}}/{\overline {WV}}}$

The equation given above can be proven using Menelaus' theorem.

Conversely, it can be concluded from the correctness of this equation that the straight lines , and intersect at a point or are parallel. This inversion of Ceva's theorem is often used in triangle geometry for evidence from the " Distinguished Points in Triangle " topic . ${\displaystyle AD}$ ${\displaystyle BE}$ ${\displaystyle CF}$

If the equation holds, it also follows:

${\displaystyle {\overline {AF}}\cdot {\overline {BD}}\cdot {\overline {CE}}={\overline {AE}}\cdot {\overline {BF}}\cdot {\overline {CD}}}$

Since orientation is lost here, this equation is not sufficient to reverse the theorem, cf. Menelaus theorem .

A generalization of Ceva's theorem is Routh's theorem .

If one formulates Ceva's theorem for the real projective plane or for the projective closure of the (affine) real intuition plane used here, one can formulate the theorem and its inversion without the special case of the parallel straight line.

literature

• Max Koecher , Aloys Krieg : level geometry . 3rd edition. Springer-Verlag, Berlin 2007, ISBN 978-3-540-49327-3 , pp. 78-81
• Branko Grunbaum, GC Shephard: Ceva, Menelaus, and the Area Principle . Mathematics Magazine, Vol. 68, No. 4 (Oct. 1995), pp. 254-268 ( JSTOR 2690569 )
• James E. Lightner: A New Look at Centers of a Triangle . The Mathematics Teacher, Volume 68, No. 7 (NOVEMBER 1975), pp. 612-615 ( JSTOR 27960289 )

Web links

Commons : Ceva's theorem  - album with pictures, videos and audio files

• Ceva's theorem on cut-the-knot.org
• Jürgen Richter-Gebert: Meditations on Ceva's Theorem
• Paul Yiu: Euclidean Geometry Notes . Script, Florida Atlantic University, pp. 87-102