Theorem of complementary parallelograms

The theorem of supplementary parallelograms is a theorem of elementary geometry about the area of parallelograms .

Formulation of the sentence

Parallelogram with set of supplementary parallelograms ${\displaystyle {\mathcal {P}}=ABCD}$
${\displaystyle F_{{\mathcal {P}}_{B}}=F_{{\mathcal {P}}_{D}}}$

The sentence says the following:

Given a parallelogram of the Euclidean plane and in it one of the two diagonals , for example ( oBdA ) . ${\displaystyle {\mathcal {P}}=ABCD}$ ${\displaystyle d}$ ${\displaystyle d=AC}$

Next is an inner point of . ${\displaystyle Z}$ ${\displaystyle d}$

Let the two parallels be drawn through to the sides of , which divide into four partial parallelograms, their sole common point being. ${\displaystyle Z}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle Z}$

Then:

The two partial parallelograms, which are not broken down by the diagonal , that is to say they only have the point in common, are complementary and therefore have an identical surface area . ${\displaystyle d}$ ${\displaystyle d}$ ${\displaystyle Z}$

Derivation and explanation

The four partial parallelograms of are denoted by. The indexing is based on the corner points of . So it is the partial parallelogram that contains the corner point     . Consequently, for reasons of convexity, the two partial parallelograms which have   only the point in common with the diagonal   , and , while and are those two partial parallelograms which have more than one point in common. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}_{A},{\mathcal {P}}_{B},{\mathcal {P}}_{C},{\mathcal {P}}_{D}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}_{X}}$ ${\displaystyle X}$   ${\displaystyle (X=A,B,C,D)}$ ${\displaystyle d}$ ${\displaystyle Z}$ ${\displaystyle {\mathcal {P}}_{B}}$ ${\displaystyle {\mathcal {P}}_{D}}$ ${\displaystyle {\mathcal {P}}_{A}}$ ${\displaystyle {\mathcal {P}}_{C}}$ ${\displaystyle d}$

${\displaystyle d}$   divided now into two congruent triangles , namely in and , and just as divided both and each congruent into two triangles. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle ABC}$ ${\displaystyle ACD}$   ${\displaystyle d}$ ${\displaystyle {\mathcal {P}}_{A}}$ ${\displaystyle {\mathcal {P}}_{C}}$

Here are now and the two splitting triangles respectively and the two splitting triangles while and within the triangle , respectively , and within the triangle area, it is in the three patches and and broken and just as in the three patches and and  . ${\displaystyle {\Delta }_{1}}$ ${\displaystyle {\Delta }_{2}}$ ${\displaystyle {\mathcal {P}}_{A}}$ ${\displaystyle {\Delta }_{3}}$ ${\displaystyle {\Delta }_{4}}$ ${\displaystyle {\mathcal {P}}_{C}}$ ${\displaystyle {\Delta }_{1}}$ ${\displaystyle {\Delta }_{3}}$ ${\displaystyle ABC}$ ${\displaystyle {\Delta }_{2}}$ ${\displaystyle {\Delta }_{4}}$ ${\displaystyle ACD}$ ${\displaystyle ABC}$ ${\displaystyle {\Delta }_{1}}$ ${\displaystyle {\Delta }_{3}}$ ${\displaystyle {\mathcal {P}}_{B}}$ ${\displaystyle ACD}$ ${\displaystyle {\Delta }_{2}}$ ${\displaystyle {\Delta }_{4}}$ ${\displaystyle {\mathcal {P}}_{D}}$

Consequently, the identities result with regard to the surface area

(I)   ${\displaystyle F_{ABC}=F_{ACD}={\frac {F_{ABCD}}{2}}}$

(II)   ${\displaystyle F_{ABC}=F_{{\Delta }_{1}}+F_{{\Delta }_{3}}+F_{{\mathcal {P}}_{B}}}$

(III)   ${\displaystyle F_{ACD}=F_{{\Delta }_{2}}+F_{{\Delta }_{4}}+F_{{\mathcal {P}}_{D}}}$

and from this, because of the aforementioned congruence relationships, directly the identity

(IV)   . ${\displaystyle F_{{\mathcal {P}}_{B}}=F_{{\mathcal {P}}_{D}}}$

This also means:

${\displaystyle {\mathcal {P}}_{B}}$   and   are complementary . ${\displaystyle {\mathcal {P}}_{D}}$

Because by adding a finite number of pairwise congruent polygons from   and   two congruent polygons are obtained ${\displaystyle {\mathcal {P}}_{B}}$ ${\displaystyle {\mathcal {P}}_{D}}$ , namely the two triangles     and   ${\displaystyle ABC}$ ${\displaystyle ACD}$

This proves the theorem.

About the terminology

The two partial parallelograms   and   are called supplementary parallelograms because of the facts presented in the sentence . This also explains the name of the sentence itself. ${\displaystyle {\mathcal {P}}_{B}}$ ${\displaystyle {\mathcal {P}}_{D}}$

literature

• PS Alexandroff , AI Markuschewitsch , AJ Chintschin [Red.]: Encyclopedia of Elementary Mathematics. Volume V. Geometry (=  university books for mathematics . Volume 11 ). German Science Publishers, Berlin 1971.
• Hermann Athens, Jörn Bruhn (ed.): Lexicon of school mathematics and related areas . tape 1 . A-E. Aulis Verlag Deubner, Cologne 1977, ISBN 3-7614-0242-2 .
• Walter Gellert, Herbert Kästner , Siegfried Neuber (Hrsg.): Fachlexikon ABC Mathematik . Verlag Harri Deutsch, Thun / Frankfurt / Main 1978, ISBN 3-87144-336-0 .
• Hugo Fenkner, Karl Holzmüller: Mathematical teaching work . According to the guidelines for the curricula of the higher schools in Prussia, revised by Karl Holzmüller. 12th edition. Geometry. Edition A in 2 parts. I. part. Published by Otto Salle, Berlin 1926.
• Johannes Kratz: Geometry (=  mathematics for high schools . Volume 4 ). 4th edition. Bayerischer Schulbuch Verlag, Munich 1966.
• Theophil Lambacher , Wilhelm Schweizer (Ed.): Lambacher-Schweizer . Mathematical teaching material for higher schools . Geometry. Edition E. Part 1. 15th edition. Ernst Klett Verlag, Stuttgart 1965.
• Harald Scheid (Ed.): DUDEN: Rechnen und Mathematik . 4th, completely revised edition. Bibliographical Institute, Mannheim / Vienna / Zurich 1985, ISBN 3-411-02423-2 .

Individual evidence

1. ↑ Hugo Fenkner, Karl Holz Müller: Mathematical Unterrichtswerk . According to the guidelines for the curricula of the higher schools in Prussia, revised by Karl Holzmüller. 12th edition. Geometry. Edition A in 2 parts. I. part. Verlag von Otto Salle, Berlin 1926, p. 127 .
2. ^ Johannes Kratz: Geometry (=  mathematics for high schools . Volume 4 ). 4th edition. Bayerischer Schulbuch Verlag, Munich 1966, p. 172 .
3. ^ Theophil Lambacher , Wilhelm Schweizer (ed.): Lambacher-Schweizer . Mathematical teaching material for higher schools . Geometry. Edition E. Part 1. 15th edition. Ernst Klett Verlag, Stuttgart 1965, p. 59 .
4. ^ Technical lexicon ABC Mathematics . S. 135 .
5. ↑ DUDEN: arithmetic and mathematics . S. 142 .
6. ^ Lexicon of school mathematics . tape 1 , p. 247 .
7. ↑ Encyclopedia of Elementary Mathematics . tape V , S. 140 .