Gauss's theorem on the complete four-sided
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.
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
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 .
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 .
The formation of the opposite corner is an involutorial mapping on :
- .
The vertex set can therefore be written as follows:
- With
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 .
The level of the full four-sided
The connecting space is an affine plane inside that contains the entire set of vertices :
This is immediately clear to the corners . Because contains but then the straight and finally .
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 .
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 :
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:
and
and is thereby clearly determined.
Gauss's theorem deals with these three midpoints of the diagonals of the complete quadrilateral .
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 .
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 .
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
- ↑ G. Bol: Elements of Analytical Geometry . 1st part, 1948, p. 28 .
- ↑ R. Brandl: Lectures on Analytical Geometry . 1996, p. 36 .
- ↑ R. Brandl: Lectures on Analytical Geometry . 1996, p. 34-38 .
- ↑ HF Baker: An Introduction to Plane Geometry . 1971, p. 11 .
- ↑ CF Adler: Modern geometry: an integrated first course . 1967, p. 143 .
- ↑ CA Scott: Projective methods in plane analytical geometry . 1961, p. 41 .
- ↑ G. Bol: Elements of Analytical Geometry . 1st part, 1948, p. 27 .
- ↑ R. Brandl: Lectures on Analytical Geometry . 1996, p. 36 .
- ↑ R. Brandl: Lectures on Analytical Geometry . 1996, p. 36 .
- ↑ G. Bol: Elements of Analytical Geometry . 1st part, 1948, p. 28 .
- ↑ R. Brandl: Lectures on Analytical Geometry . 1996, p. 36 .
- ↑ M. Koecher, A. Krieg: level geometry . 2000, p. 64 .
- ↑ R. Brandl: Lectures on Analytical Geometry . 1996, p. 36 .
- ↑ 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.