Gauss's theorem on the complete four-sided

4 sides (black), 3 diagonals (blue), common straight line of the diagonal centers (red)

The Gauss on the complete quadrilateral is a set of affine geometry . It goes back to Carl Friedrich Gauß (1777–1855), who found it in 1810. The sentence belongs to the series of so-called closing sentences, which also include the sentence of Pappos-Pascal , the sentence of Desargues , the sentence of Menelaus and the sentence of Ceva .

Clarification of the terms

An affine space over a body is given . A complete quadrilateral in (English. Sometimes as a quadrilateral or rather as complete quadrilateral called) consists of four different lines , which pairs cut , none of which three by a single point of going. ${\displaystyle {\mathcal {A}}}$ ${\displaystyle K}$ ${\displaystyle 2=1+1\neq 0}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle g_{0},\,g_{1},\,g_{2},\,g_{3}}$ ${\displaystyle {\mathcal {A}}}$

The corners of the full four-sided

The paired intersections of the four starting straight lines are called the corners of the complete quadrilateral and form the set of corners . Thereby, to every 2-set of straight lines, the corner of what is in total belongs to ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {E}}={\mathcal {E}}({\mathcal {Q}})}$ ${\displaystyle {\{g_{k},\,g_{l}\}}_{\neq }\subset {\{g_{0},\,g_{1},\,g_{2},\,g_{3}\}}}$ ${\displaystyle E=g_{k}\cap g_{l}}$ ${\displaystyle {\mathcal {Q}}}$

${\displaystyle |{\mathcal {E}}|={\binom {4}{2}}=6}$

${\displaystyle {\mathcal {Q}}}$ Corners leads.

Furthermore, there are exactly three corners on each straight line , namely those corners which arise as points of intersection with the other straight lines . ${\displaystyle g_{k}(k=0,\,1,\,2,\,3)}$ ${\displaystyle g_{k}\cap g_{j}}$ ${\displaystyle g_{k}}$ ${\displaystyle g_{j}(j\neq k)}$

In addition, associated with each corner biunique the opposite corner or complementary corner which one, characterized wins that complement the corresponding forms and then to opposite vertex associated as . ${\displaystyle E=g_{k}\cap g_{l}({\{g_{k},\,g_{l}\}}_{\neq }\subset {\{g_{0},\,g_{1},\,g_{2},\,g_{3}\}})}$ ${\displaystyle E^{'}}$ ${\displaystyle {\{g_{0},\,g_{1},\,g_{2},\,g_{3}\}}\setminus {\{g_{k},\,g_{l}\}}={\{g_{m},\,g_{n}\}}}$ ${\displaystyle E}$ ${\displaystyle E^{'}=g_{m}\cap g_{n}}$

The formation of the opposite corner is an involutorial mapping on : ${\displaystyle {\mathcal {E}}}$

${\displaystyle E^{''}=E}$   ${\displaystyle (E\in {\mathcal {E}})}$.

The vertex set can therefore be written as follows: ${\displaystyle {\mathcal {E}}}$

${\displaystyle {\mathcal {E}}=\{E_{1},\,{E_{1}}^{'},\,E_{2},\,{E_{2}}^{'},\,E_{3},\,{E_{3}}^{'}\}}$ With

${\displaystyle E_{1}=g_{0}\cap g_{1}}$   ${\displaystyle {E_{1}}^{'}=g_{2}\cap g_{3}}$

${\displaystyle E_{2}=g_{0}\cap g_{2}}$   ${\displaystyle {E_{2}}^{'}=g_{1}\cap g_{3}}$

${\displaystyle E_{3}=g_{0}\cap g_{3}}$   ${\displaystyle {E_{3}}^{'}=g_{1}\cap g_{2}}$

If one carries out this consideration with one of the three of the different straight lines instead of with , one obtains a correspondingly different but equivalent representation of the set of corners . The relationship between corners and opposite corners is unaffected by the type of representation of the set of corners and depends solely on that of the four starting straight lines . ${\displaystyle g_{0}}$ ${\displaystyle g_{0}}$ ${\displaystyle {\mathcal {E}}={\mathcal {E}}({\mathcal {Q}})}$

The level of the full four-sided

The connecting space is an affine plane inside that contains the entire set of vertices : ${\displaystyle {\mathcal {P}}=g_{0}\lor g_{1}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {E}}}$

This is immediately clear to the corners . Because contains but then the straight and finally . ${\displaystyle E_{1},\,E_{2},\,E_{3},\,{E_{2}}^{'},\,{E_{3}}^{'}}$ ${\displaystyle g_{2}=E_{2}\lor {E_{3}}^{'}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle g_{2}}$ ${\displaystyle {E_{1}}^{'}}$

${\displaystyle {\mathcal {P}}}$is therefore independent of the type of representation of the set of corners, the plane belonging to the complete quadrilateral and generated by it within . ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {\mathcal {P}}={\mathcal {P}}({\mathcal {Q}})}$ ${\displaystyle {\mathcal {A}}}$

The diagonals of the complete quadrilateral and their midpoints

According to the construction , the two corners and at the same time for no index   lie on one of the four given straight lines . If you connect every corner of with the opposite corner , you get three more straight lines in addition to the four given straight lines . These are the diagonals of the full four-sided : ${\displaystyle i=1,\,2,\,3}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle E_{i}}$ ${\displaystyle {E_{i}}^{'}}$ ${\displaystyle g_{0},\,g_{1},\,g_{2},\,g_{3}}$ ${\displaystyle E_{i}}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle {E_{i}}^{'}}$ ${\displaystyle d_{1},\,d_{2},\,d_{3}}$ ${\displaystyle {\mathcal {Q}}}$

${\displaystyle d_{1}=E_{1}\lor {E_{1}}^{'}}$

${\displaystyle d_{2}=E_{2}\lor {E_{2}}^{'}}$

${\displaystyle d_{3}=E_{3}\lor {E_{3}}^{'}}$

For each of the three diagonals, there is a marked point under the points which incise with . This point is called the center of the diagonal or, for short, the center of the diagonal . The center of the diagonal satisfies the equations: ${\displaystyle d_{i}(i=1,\,2,\,3)}$ ${\displaystyle d_{i}}$ ${\displaystyle M_{i}}$ ${\displaystyle d_{i}}$ ${\displaystyle d_{i}}$ ${\displaystyle d_{i}}$

${\displaystyle {\overrightarrow {{E_{i}}{M_{i}}}}={\tfrac {1}{2}}\cdot {\overrightarrow {{E_{i}}{E_{i}}^{'}}}={\overrightarrow {{M_{i}}{{E_{i}}^{'}}}}}$

and

${\displaystyle E_{i}+{\tfrac {1}{2}}\cdot {\overrightarrow {{E_{i}}{E_{i}}^{'}}}=M_{i}={E_{i}}^{'}-{\tfrac {1}{2}}\cdot {\overrightarrow {{E_{i}}{E_{i}}^{'}}}}$

and is thereby clearly determined.

Gauss's theorem deals with these three midpoints of the diagonals of the complete quadrilateral ${\displaystyle {\mathcal {Q}}}$ .

formulation

The sentence is as follows:

In an affine space over a body of the characteristic , the center points of the diagonals of a complete quadrilateral always lie on a straight line, the so-called Gaussian line . ${\displaystyle K}$ ${\displaystyle \operatorname {char} (K)\neq 2}$

The fall of the Euclidean plane

The theorem applies in particular to the case that the coordinate plane is above . A particularly noteworthy case is if is, i.e. the field of real numbers is present and if the given affine space then coincides with the Euclidean plane . ${\displaystyle {\mathcal {A}}={\mathbb {A} }_{2}(K)}$ ${\displaystyle K}$ ${\displaystyle K=\mathbb {R} }$ ${\displaystyle {\mathcal {A}}}$

Under these circumstances, the sentence can then be pronounced like this:

If four straight lines lie in the Euclidean plane in such a way that no three of them pass through a point, the centers of the associated diagonals always lie on a straight line .

literature

• Claire Fisher Adler: Modern geometry: an integrated first course . 2nd Edition. McGraw-Hill , New York (et al.) 1967.
• HF Baker : An Introduction to Plane Geometry . Reprint. Chelsea Publishing Company , Bronx, NY 1971, ISBN 0-8284-0247-7 .
• Gerrit Bol : Elements of Analytical Geometry. 1st part . Vandenhoeck & Ruprecht , Göttingen 1948.
• Rolf Brandl: Lectures on analytical geometry . Publisher Rolf Brandl, Hof 1996.
• Max Koecher , Aloys Krieg : level geometry (=  Springer textbook ). 2nd, revised and expanded edition. Springer Verlag , Berlin (among others) 2000, ISBN 3-540-67643-0 .
• Charlotte Angas Scott: Projective methods in plane analytical geometry . 3. Edition. Chelsea Publishing Company, New York, NY 1961, ISBN 0-8284-0146-2 .

Individual evidence

1. ↑ G. Bol: Elements of Analytical Geometry . 1st part, 1948, p. 28 .
2. ↑ R. Brandl: Lectures on Analytical Geometry . 1996, p. 36 .
3. ↑ R. Brandl: Lectures on Analytical Geometry . 1996, p. 34-38 .
4. ↑ HF Baker: An Introduction to Plane Geometry . 1971, p. 11 .
5. ↑ CF Adler: Modern geometry: an integrated first course . 1967, p. 143 .
6. ↑ CA Scott: Projective methods in plane analytical geometry . 1961, p. 41 .
7. ↑ G. Bol: Elements of Analytical Geometry . 1st part, 1948, p. 27 .
8. ↑ R. Brandl: Lectures on Analytical Geometry . 1996, p. 36 .
9. ↑ R. Brandl: Lectures on Analytical Geometry . 1996, p. 36 .
10. ↑ G. Bol: Elements of Analytical Geometry . 1st part, 1948, p. 28 .
11. ↑ R. Brandl: Lectures on Analytical Geometry . 1996, p. 36 .
12. ↑ M. Koecher, A. Krieg: level geometry . 2000, p. 64 .
13. ↑ R. Brandl: Lectures on Analytical Geometry . 1996, p. 36 .
14. ↑ This representation follows on from that of Gerrit Bol ( Elements of Analytical Geometry . Part 1, 1948, pp. 27–28) and builds a bridge to the above sketch.