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In mathematics, **Pappus's hexagon theorem** (attributed to Pappus of Alexandria) states that

**Pappus of Alexandria** was one of the last great Greek mathematicians of Antiquity, known for his *Synagoge* (Συναγωγή) or *Collection*, and for Pappus's hexagon theorem in projective geometry. Nothing is known of his life, other than, that he had a son named Hermodorus, and was a teacher in Alexandria.

- Proof: affine form
- Proof with homogeneous coordinates
- Dual theorem
- Other statements of the theorem
- Origins
- Notes
- References
- External links

- given one set of collinear points , and another set of collinear points , then the intersection points of line pairs and and and are collinear, lying on the
*Pappus line*. These three points are the points of intersection of the "opposite" sides of the hexagon .

In geometry, **collinearity** of a set of points is the property of their lying on a single line. A set of points with this property is said to be **collinear**. In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".

It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring.^{ [1] } Projective planes in which the "theorem" is valid are called **pappian planes**.

In mathematics, a **projective plane** is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus *any* two distinct lines in a projective plane intersect in one and only one point.

In abstract algebra, a **division ring**, also called a **skew field**, is a ring in which division is possible. Specifically, it is a nonzero ring in which every nonzero element a has a multiplicative inverse, i.e., an element x with *a*·*x* = *x*·*a* = 1. Stated differently, a ring is a division ring if and only if the group of units equals the set of all nonzero elements. A division ring is a type of noncommutative ring under the looser definition where *noncommutative ring* refers to rings which are not *necessarily* commutative.

If one restricts the projective plane such that the Pappus line is the line at infinity, one gets the *affine version* of Pappus's theorem shown in the second diagram.

If the Pappus line and the lines have a point in common, one gets the so called **little** version of Pappus's theorem^{ [2] }.

The dual of this incidence theorem states that given one set of concurrent lines , and another set of concurrent lines , then the lines defined by pairs of points resulting from pairs of intersections and and and are concurrent. (*Concurrent* means that the lines pass through one point.)

In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) **duality** is the formalization of this concept. There are two approaches to the subject of duality, one through language and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a * duality*. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry.

In projective geometry, an **intersection theorem** or **incidence theorem** is a statement concerning an incidence structure – consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of objects *A* and *B*. The "theorem" states that, whenever a set of objects satisfies the incidences, then the objects *A* and *B* must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is a property that some geometries satisfy but others don't.

In geometry, three or more lines in a plane or higher-dimensional space are said to be **concurrent** if they intersect at a single point.

Pappus's theorem is a special case of Pascal's theorem for a conic—the limiting case when the conic degenerates into 2 straight lines. Pascal's theorem is in turn a special case of the Cayley–Bacharach theorem.

In logic, especially as applied in mathematics, concept A is a **special case** or specialization of concept B precisely if every instance of A is also an instance of B but not vice versa, or equivalently, if B is a generalization of A. A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case. A degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed.

In projective geometry, **Pascal's theorem** states that if six arbitrary points are chosen on a conic and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon meet at three points which lie on a straight line, called the **Pascal line** of the hexagon. It is named after Blaise Pascal.

In mathematics, a **limiting case** of a mathematical object is a special case that arises when one or more components of the object take on their most extreme possible values. For example:

The Pappus configuration is the configuration of 9 lines and 9 points that occurs in Pappus's theorem, with each line meeting 3 of the points and each point meeting 3 lines. In general, the Pappus line does not pass through the point of intersection of and .^{ [3] } This configuration is self dual. Since, in particular, the lines have the properties of the lines of the dual theorem, and collinearity of is equivalent to concurrence of , the dual theorem is therefore just the same as the theorem itself. The Levi graph of the Pappus configuration is the Pappus graph, a bipartite distance-regular graph with 18 vertices and 27 edges.

The projective form of Pappus's theorem is proven, if the affine form of the statement can be proven (the extension of a pappian plane to a projective plane is unique !).

Because of the parallelity in an affine plane one has to distinct two cases: and . The key for a simple proof is the possibility for introducing a "suitable" coordinate system:

**Case 1:** The lines intersect at point .

In this case coordinates are introduced, such that (see diagram). have the coordinates .

From the parallelity of the lines one gets and the parallelity of the lines yields . Hence line has slope and is parallel line .

**Case 2:** (little theorem).

In this case the coordinates are chosen such that . From the parallelity of and one gets and , respectively, and at least the parallelity .

Choose homogeneous coordinates with

- .

On the lines , given by , take the points to be

for some . The three lines are , so they pass through the same point if and only if . The condition for the three lines and with equations to pass through the same point is . So this last set of three lines is concurrent if all the other eight sets are because multiplication is commutative, so . Equivalently, are collinear.

The proof above also shows that for Pappus's theorem to hold for a projective space over a division ring it is both sufficient and necessary that the division ring is a (commutative) field. German mathematician Gerhard Hessenberg proved that Pappus's theorem implies Desargues's theorem.^{ [4] }^{ [5] } In general, Pappus's theorem holds for some projective plane if and only if it is a projective plane over a commutative field. The projective planes in which Pappus's theorem does not hold are Desarguesian projective planes over noncommutative division rings, and non-Desarguesian planes.

The proof is invalid if happen to be collinear. In that case an alternative proof can be provided, for example, using a different projective reference.

Because of the principle of duality for projective planes the **dual theorem of Pappus** is true:

If 6 lines are chosen alternately from two pencils with centers , the lines

are concurrent, that means: they have a point in common.

The left diagram shows the projective version, the right one an affine version, where the points are points at infinity. If point is on the line than one gets the "dual little theorem" of Ppappus' theorem.

- dual theorem: projective form
- dual theorem: affine form

If in the affine version of the dual "little theorem" point is a point at infinity too, one gets Thomsen's theorem, a statement on 6 points on the sides of a triangle (see diagram). The Thomsen figure plays an essential role coordinatising an axiomatic defined projective plane^{ [6] }. The proof of the closure of Thomsen's figure is covered by the proof for the "little theorem", given above. But there exists a simple direct proof, too:

Because the statement of Thomsen's theorem (the closure of the figure) uses only the terms *connect, intersect* and *parallel*, the statement is affinely invariant, and one can introduce coordinates such that (see right diagram). The starting point of the sequence of chords is One easily verifies the coordinates of the points given in the diagram, which shows: the last point coincides with the first popint.

*Thomsen figure*(points of the triangle ) as dual theorem of the little theorem of Pappus ( is at infinity, too !).- Thomsen figure: proof

In addition to the above characterizations of Pappus's theorem and its dual, the following are equivalent statements:

- If the six vertices of a hexagon lie alternately on two lines, then the three points of intersection of pairs of opposite sides are collinear.
^{ [7] } - Arranged in a matrix of nine points (as in the figure and description above) and thought of as evaluating a permanent, if the first two rows and the six "diagonal" triads are collinear, then the third row is collinear.

- That is, if are lines, then Pappus's theorem states that must be a line. Also, note that the same matrix formulation applies to the dual form of the theorem when
*etc.*are triples of concurrent lines.^{ [8] }

- Given three distinct points on each of two distinct lines, pair each point on one of the lines with one from the other line, then the joins of points not paired will meet in (opposite) pairs at points along a line.
^{ [9] } - If two triangles are perspective in at least two different ways, then they are perspective in three ways.
^{ [4] } - If and are concurrent and and are concurrent, then and are concurrent.
^{ [8] }

In its earliest known form, Pappus's Theorem is Propositions 138, 139, 141, and 143 of Book VII of Pappus's *Collection*.^{ [10] } These are Lemmas XII, XIII, XV, and XVII in the part of Book VII consisting of lemmas to the first of the three books of Euclid's *Porisms.*

The lemmas are proved in terms of what today is known as the cross ratio of four collinear points. Three earlier lemmas are used. The first of these, Lemma III, has the diagram below (which uses Pappus's lettering, with G for Γ, D for Δ, J for Θ, and L for Λ).

Here three concurrent straight lines, AB, AG, and AD, are crossed by two lines, JB and JE, which concur at J. Also KL is drawn parallel to AZ. Then

- KJ : JL :: (KJ : AG & AG : JL) :: (JD : GD & BG : JB).

These proportions might be written today as equations:^{ [11] }

- KJ/JL = (KJ/AG)(AG/JL) = (JD/GD)(BG/JB).

The last compound ratio (namely JD : GD & BG : JB) is what is known today as the cross ratio of the collinear points J, G, D, and B in that order; it is denoted today by (J, G; D, B). So we have shown that this is independent of the choice of the particular straight line JD that crosses the three straight lines that concur at A. In particular

- (J, G; D, B) = (J, Z; H, E).

It does not matter on which side of A the straight line JE falls. In particular, the situation may be as in the next diagram, which is the diagram for Lemma X.

Just as before, we have (J, G; D, B) = (J, Z; H, E). Pappus does not explicitly prove this; but Lemma X is a converse, namely that if these two cross ratios are the same, and the straight lines BE and DH cross at A, then the points G, A, and Z must be collinear.

What we showed originally can be written as (J, ∞; K, L) = (J, G; D, B), with ∞ taking the place of the (nonexistent) intersection of JK and AG. Pappus shows this, in effect, in Lemma XI, whose diagram, however, has different lettering:

What Pappus shows is DE.ZH : EZ.HD :: GB : BE, which we may write as

- (D, Z; E, H) = (∞, B; E, G).

The diagram for Lemma XII is:

The diagram for Lemma XIII is the same, but BA and DG, extended, meet at N. In any case, considering straight lines through G as cut by the three straight lines through A, (and accepting that equations of cross ratios remain valid after permutation of the entries,) we have by Lemma III or XI

- (G, J; E, H) = (G, D; ∞ Z).

Considering straight lines through D as cut by the three straight lines through B, we have

- (L, D; E, K) = (G, D; ∞ Z).

Thus (E, H; J, G) = (E, K; D, L), so by Lemma X, the points H, M, and K are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon ADEGBZ are collinear.

Lemmas XV and XVII are that, if the point M is determined as the intersection of HK and BG, then the points A, M, and D are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon BEKHZG are collinear.

- ↑ Coxeter, pp. 236–7
- ↑ Rolf Lingenberg:
*Grundlagen der Geometrie*, BI-Taschenbuch, 1969, p. 93 - ↑ However, this does occur when and are in perspective, that is, and are concurrent.
- 1 2 Coxeter 1969 , p. 238
- ↑ According to ( Dembowski 1968 , pg. 159, footnote 1), Hessenberg's original proof Hessenberg (1905) is not complete; he disregarded the possibility that some additional incidences could occur in the Desargues configuration. A complete proof is provided by Cronheim 1953.
- ↑ W. Blaschke:
*Projektive Geometrie*, Springer-Verlag, 2013, ISBN 3034869320, S. 190 - ↑ Coxeter, p. 231
- 1 2 Coxeter, p. 233
- ↑ Whicher, chapter 14
- ↑ Heath (Vol. II, p. 421) cites these propositions. The latter two can be understood as converses of the former two. Kline (p. 128) cites only Proposition 139. The numbering of the propositions is as assigned by Hultsch.
- ↑ A reason for using the notation above is that, for the ancient Greeks, a ratio is not a number or a geometrical object. We may think of ratio today as an equivalence class of pairs of geometrical objects. Also, equality for the Greeks is what we might today call congruence. In particular, distinct line segments may be equal. Ratios are not
*equal*in this sense; but they may be the*same.*

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**Projective geometry** is a topic in mathematics. It is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points to Euclidean points, and vice versa.

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In mathematics, a **affine algebraic plane curve** is the zero set of a polynomial in two variables. A **projective algebraic plane curve** is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve can be restricted to an affine algebraic plane curve by replacing by one some indeterminate of the defining homogeneous polynomial. As these two operations are each inverse to the other, the phrase **algebraic plane curve** is often used without specifying explicitly whether it is the affine or the projective case that is considered.

In mathematics, the **upper half-plane****H** is the set of points (*x*, *y*) in the Cartesian plane with *y* > 0.

In projective geometry, **Desargues's theorem**, named after Girard Desargues, states:

In geometry, a **degenerate conic** is a conic that fails to be an irreducible curve. This means that the defining equation is factorable over the complex numbers as the product of two linear polynomials.

In geometry, the **cross-ratio**, also called the **double ratio** and **anharmonic ratio**, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points *A*, *B*, *C* and *D* on a line, their cross ratio is defined as

The notion of **line** or **straight line** was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects. Until the 17th century, lines were defined as the "[…] first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width. […] The straight line is that which is equally extended between its points."

In projective geometry, a **homography** is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include **projectivity**, **projective transformation**, and **projective collineation**.

In mathematics, a **Minkowski plane** is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane.

In Euclidean and projective geometry, just as two (distinct) points determine a line, **five points determine a conic**. There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines.

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In mathematics, a **Möbius plane** is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane. The classical example is based on the geometry of lines and circles in the real affine plane.

In geometry, specifically projective geometry, a **blocking set** is a set of points in a projective plane which every line intersects and which does not contain an entire line. The concept can be generalized in several ways. Instead of talking about points and lines, one could deal with *n*-dimensional subspaces and *m*-dimensional subspaces, or even more generally, objects of type 1 and objects of type 2 when some concept of intersection makes sense for these objects. A second way to generalize would be to move into more abstract settings than projective geometry. One can define a blocking set of a hypergraph as a set that meets all edges of the hypergraph.

- Coxeter, Harold Scott MacDonald (1969),
*Introduction to Geometry*(2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50458-0, MR 0123930 - Cronheim, A. (1953), "A proof of Hessenberg's theorem",
*Proceedings of the American Mathematical Society*,**4**: 219–221, doi:10.2307/2031794 - Dembowski, Peter (1968),
*Finite Geometries*, Berlin: Springer Verlag - Heath, Thomas (1981) [1921],
*A History of Greek Mathematics*, New York: Dover - Hessenberg, Gerhard (1905), "Beweis des Desarguesschen Satzes aus dem Pascalschen",
*Mathematische Annalen*, Berlin / Heidelberg: Springer,**61**(2): 161–172, doi:10.1007/BF01457558, ISSN 1432-1807 - Hultsch, Fridericus (1877),
*Pappi Alexandrini Collectionis Quae Supersunt*, Berlin - Kline, Morris (1972),
*Mathematical Thought From Ancient to Modern Times*, New York: Oxford University Press - Whicher, Olive (1971),
*Projective Geometry*, Rudolph Steiner Press, ISBN 0-85440-245-4

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