Diagonal set

The diagonal theorem is a theorem of elementary geometry with which a characteristic condition is formulated, under which a rectangle of the Euclidean plane is a parallelogram .

Formulation of the sentence

Parallelogram with diagonals

The sentence says the following:

Given a rectangle of the Euclidean plane. ${\displaystyle \square {ABCD}}$

Then:

${\displaystyle \square {ABCD}}$is in any case a parallelogram when the two diagonals and bisect each other in such a way that the centers of the two diagonals coincide. ${\displaystyle {\overline {AC}}}$ ${\displaystyle {\overline {BD}}}$

Derivation by means of vector calculation

The condition says that there is a point in the Euclidean plane such that the two vector equations and exist. ${\displaystyle S}$ ${\displaystyle {\overrightarrow {AS}}={\overrightarrow {SC}}}$ ${\displaystyle {\overrightarrow {BS}}={\overrightarrow {SD}}}$

From this one deduces:

${\displaystyle {\overrightarrow {AD}}={\overrightarrow {AS}}+{\overrightarrow {SD}}={\overrightarrow {SC}}+{\overrightarrow {BS}}={\overrightarrow {BS}}+{\overrightarrow {SC}}={\overrightarrow {BC}}}$.

In the same way:

${\displaystyle {\overrightarrow {AB}}={\overrightarrow {AS}}+{\overrightarrow {SB}}={(-{\overrightarrow {CS}})}+{(-{\overrightarrow {BS}})}={(-{\overrightarrow {CS}})}+{(-{\overrightarrow {SD}})}=-({\overrightarrow {CS}}+{\overrightarrow {SD}})=-{\overrightarrow {CD}}={\overrightarrow {DC}}}$.

This proves the theorem.

Generalization on coordinate planes

The diagonal theorem can be extended and sharpened to affine coordinate planes over commutative fields of a characteristic ; as follows: ${\displaystyle {\mathbb {A} }_{2}(K)=K^{2}}$ ${\displaystyle K}$ ${\displaystyle \operatorname {char} (K)\neq 2}$

Given are four non-collinear points that differ in pairs . ${\displaystyle A,B,C,D\in {\mathbb {A} }_{2}(K)}$

Then the following two conditions are equivalent:

(A1) The four points form a parallelogram; ie:

There are and . ${\displaystyle {A\vee B}\parallel {C\vee D}}$ ${\displaystyle {A\vee D}\parallel {B\vee C}}$

(A2) The two diagonals and intersect at the center of the two diagonals; ie: ${\displaystyle {A\vee C}}$ ${\displaystyle {B\vee D}}$

It applies . ${\displaystyle {\frac {1}{2}}(A+C)={\frac {1}{2}}(B+D)}$

Comment on coordinate planes over bodies of characteristic 2

The situation is different for a commutative field of the characteristic . If in this case four points form a parallelogram, the diagonals are parallel. ${\displaystyle K}$ ${\displaystyle \operatorname {char} (K)=2}$

See also

• Varignon set
• Commandino's theorem

literature

• Max Koecher , Aloys Krieg : level geometry (=  Springer textbook ). 2nd, revised and expanded edition. Springer Verlag , Berlin (among others) 2000, ISBN 3-540-67643-0 .
• Harald Scheid (Ed.): DUDEN: Rechnen und Mathematik . 4th, completely revised edition. Bibliographical Institute , Mannheim - Vienna - Zurich 1985, ISBN 3-411-02423-2 .

References and footnotes

1. ↑ DUDEN: arithmetic and mathematics. 1985, p. 652
2. ↑ Koecher War: Level Geometry. 2000, p. 59
3. ↑ For two points is the connecting line . ${\displaystyle X,Y\in {\mathbb {A} }_{2}(K)}$ ${\displaystyle {X\vee Y}}$
4. ↑ Koecher War: Level Geometry. 2000, p. 60