Pappus's hexagon theorem

Pappus's hexagon theorem Pappus's hexagon theorem: Points X, Y and Z are collinear on the Pappus line. The hexagon is AbCaBc. Pappus's theorem: affine form {displaystyle Abparallel aB,Bcparallel bCRightarrow Acparallel aC} Na matemática, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of collinear points {estilo de exibição A,B,C} , and another set of collinear points {estilo de exibição a,b,c} , then the intersection points {estilo de exibição X,S,Z} of line pairs {displaystyle Ab} e {displaystyle aB,Ac} e {displaystyle aC,Bc} e {displaystyle bC} are collinear, lying on the Pappus line. These three points are the points of intersection of the "opposite" sides of the hexagon {displaystyle AbCaBc} .

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 "teorema" is valid are called pappian planes.

If one restricts the projective plane such that the Pappus line {estilo de exibição você} is the line at infinity, one gets the affine version of Pappus's theorem shown in the second diagram.

If the Pappus line {estilo de exibição você} and the lines {estilo de exibição g,h} 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 {estilo de exibição A,B,C} , and another set of concurrent lines {estilo de exibição a,b,c} , then the lines {estilo de exibição x,y,z} defined by pairs of points resulting from pairs of intersections {displaystyle Acap b} e {displaystyle acap B,;Acap c} e {displaystyle acap C,;Bcap c} e {displaystyle bcap C} are concurrent. (Concurrent means that the lines pass through one 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.

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. No geral, the Pappus line does not pass through the point of intersection of {estilo de exibição ABC} e {displaystyle abc} .[3] This configuration is self dual. Desde, em particular, the lines {displaystyle Bc,bC,XY} have the properties of the lines {estilo de exibição x,y,z} of the dual theorem, and collinearity of {estilo de exibição X,S,Z} is equivalent to concurrence of {displaystyle Bc,bC,XY} , 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 arestas.

Conteúdo 1 Prova: affine form 2 Proof with homogeneous coordinates 3 Dual theorem 4 Other statements of the theorem 5 Origins 6 Notas 7 Referências 8 External links Proof: affine form Pappus theorem: proof If the affine form of the statement can be proven, then the projective form of Pappus's theorem is proven, as 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: {displaystyle gnot parallel h} e {displaystyle gparallel h} . The key for a simple proof is the possibility for introducing a "suitable" coordinate system: Caso 1: The lines {estilo de exibição g,h} intersect at point {displaystyle S=gcap h} .

In this case coordinates are introduced, de tal modo que {estilo de exibição ;S=(0,0),;A=(0,1),;c=(1,0);} (veja o diagrama). {estilo de exibição B,C} have the coordinates {estilo de exibição ;B=(0,gama ),;C=(0,delta ),;gama ,delta notin {0,1}} .

From the parallelity of the lines {displaystyle Bc,;Cb} um fica {displaystyle b=({tfrac {delta }{gama }},0)} and the parallelity of the lines {displaystyle Ab,Ba} yields {displaystyle a=(delta ,0)} . Hence line {displaystyle Ca} has slope {estilo de exibição -1} and is parallel line {displaystyle Ac} .

Caso 2: {displaystyle gparallel h } (little theorem).

In this case the coordinates are chosen such that {estilo de exibição ;c=(0,0),;b=(1,0),;A=(0,1),;B=(gama ,1),;gamma neq 0} . From the parallelity of {displaystyle Abparallel Ba} e {displaystyle cBparallel bC} um fica {estilo de exibição ;C=(gama +1,1);} e {estilo de exibição ;a=(gama +1,0);} , respectivamente, and at least the parallelity {estilo de exibição ;Acparallel Ca;} .

Proof with homogeneous coordinates Choose homogeneous coordinates with {estilo de exibição C=(1,0,0),;c=(0,1,0),;X=(0,0,1),;A=(1,1,1)} .

On the lines {estilo de exibição AC,Ac,AX} , dado por {estilo de exibição x_{2}=x_{3},;x_{1}=x_{3},;x_{2}=x_{1}} , take the points {estilo de exibição B,S,b} to be {estilo de exibição B=(p,1,1),;Y=(1,q,1),;b=(1,1,r)} para alguns {estilo de exibição p,q,r} . The three lines {displaystyle XB,CY,cb} são {estilo de exibição x_{1}=x_{2}p,;x_{2}=x_{3}q,;x_{3}=x_{1}r} , so they pass through the same point {estilo de exibição a} se e apenas se {displaystyle rqp=1} . The condition for the three lines {displaystyle Cb,cB} e {displaystyle XY} with equations {estilo de exibição x_{2}=x_{1}q,;x_{1}=x_{3}p,;x_{3}=x_{2}r} to pass through the same point {estilo de exibição Z} é {displaystyle rpq=1} . So this last set of three lines is concurrent if all the other eight sets are because multiplication is commutative, assim {displaystyle pq=qp} . Equivalentemente, {estilo de exibição X,S,Z} 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] No geral, 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 {estilo de exibição C,c,X} happen to be collinear. In that case an alternative proof can be provided, por exemplo, using a different projective reference.

Dual theorem Because of the principle of duality for projective planes the dual theorem of Pappus is true: Se 6 lines {estilo de exibição A,b,C,uma,B,c} are chosen alternately from two pencils with centers {estilo de exibição G,H} , the lines {estilo de exibição X:=(Acap b)(acap B),} {estilo de exibição Y:=(ccap A)(Ccap a),} {estilo de exibição Z:=(bcap C)(Bcap c)} are concurrent, that means: they have a point {estilo de exibição U} in common.

The left diagram shows the projective version, the right one an affine version, where the points {estilo de exibição G,H} are points at infinity. If point {estilo de exibição U} is on the line {displaystyle GH} than one gets the "dual little theorem" of Pappus' theorem.

dual theorem: projective form dual theorem: affine form If in the affine version of the dual "little theorem" ponto {estilo de exibição U} is a point at infinity too, one gets Thomsen's theorem, a statement on 6 points on the sides of a triangle (veja o diagrama). 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, também: 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 {estilo de exibição P=(0,0),;Q=(1,0),;R=(0,1)} (see right diagram). The starting point of the sequence of chords is {estilo de exibição (0,lambda ).} One easily verifies the coordinates of the points given in the diagram, which shows: the last point coincides with the first point.

Thomsen figure (pontos {cor do estilo de exibição {vermelho}1,2,3,4,5,6} of the triangle {displaystyle PQR} ) as dual theorem of the little theorem of Pappus ( {estilo de exibição U} is at infinity, também !).

Thomsen figure: proof Other statements of the theorem Triangles {displaystyle XcC} e {displaystyle BbY} are perspective from {estilo de exibição A} e {estilo de exibição a} , e entao, also from {estilo de exibição Z} .

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. {estilo de exibição à esquerda|{começar{matriz}A&B&C\a&b&c\X&Y&Zend{matriz}}certo|} Aquilo é, E se {estilo de exibição ABC,abc,AbZ,BcX,CaY,XbC,YcA,ZaB } are lines, then Pappus's theorem states that {displaystyle XYZ} must be a line. Também, note that the same matrix formulation applies to the dual form of the theorem when {estilo de exibição (UMA,B,C)} 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] Se {estilo de exibição ;AB,CD,;} e {displaystyle EF} are concurrent and {displaystyle DE,FA,} e {estilo de exibição BC} are concurrent, então {estilo de exibição AD,SER,} e {displaystyle CF} are concurrent.[8] Origins In its earliest known form, Pappus's Theorem is Propositions 138, 139, 141, e 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. Em particular (J, G; D, B) = (J, Z; H, E).

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

Just as before, temos (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, UMA, and Z must be collinear.

What we showed originally can be written as (J, ∞; K, eu) = (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, Contudo, has different lettering: What Pappus shows is DE.ZH : EZ.HD :: GB : SER, 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, temos (eu, D; E, K) = (G, D; ∞ Z).

Desta forma (E, H; J, G) = (E, K; D, eu), so by Lemma X, the points H, M, and K are collinear. Aquilo é, 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. Aquilo é, the points of intersection of the pairs of opposite sides of the hexagon BEKHZG are collinear.

Notas ^ Coxeter, pp. 236–7 ^ Rolf Lingenberg: Grundlagen der Geometrie, BI-Taschenbuch, 1969, p. 93 ^ However, this does occur when {estilo de exibição ABC} e {displaystyle abc} are in perspective, isso é, {displaystyle Aa,Bb} e {displaystyle Cc} are concurrent. ^ Saltar para: a b Coxeter 1969, p. 238 ^ According to (Dembowski 1968, página. 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 ^ Saltar para: a b Coxeter, p. 233 ^ Whicher, capítulo 14 ^ Heath (Volume. 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. Também, equality for the Greeks is what we might today call congruence. Em particular, distinct line segments may be equal. Ratios are not equal in this sense; but they may be the same. Referências Coxeter, Harold Scott MacDonald (1969), Introdução à Geometria (2ª edição), Nova york: John Wiley & Sons, ISBN 978-0-471-50458-0, MR 0123930 Cronheim, UMA. (1953), "A proof of Hessenberg's theorem", Anais da American Mathematical Society, 4 (2): 219–221, doi:10.2307/2031794, JSTOR 2031794 Dembowski, Peter (1968), Finite Geometries, Berlim: Springer Verlag Heath, Thomas (1981) [1921], A History of Greek Mathematics, Nova york: Dover Hessenberg, Gerhard (1905), "Beweis des Desarguesschen Satzes aus dem Pascalschen", Anais Matemáticos, Berlim / Heidelberg: Springer, 61 (2): 161–172, doi:10.1007/BF01457558, ISSN 1432-1807, S2CID 120456855 Hultsch, Fridericus (1877), Pappi Alexandrini Collectionis Quae Supersunt, Berlin Kline, Morris (1972), Mathematical Thought From Ancient to Modern Times, Nova york: Oxford University Press Pambuccian, Vencedor; Schacht, Celia (2019), "The axiomatic destiny of the theorems of Pappus and Desargues", in Dani, S. G.; Papadopoulos, UMA. (ed.), Geometry in history, Springer, pp. 355–399, ISBN 978-3-030-13611-6 Whicher, Olive (1971), Projective Geometry, Rudolph Steiner Press, ISBN 0-85440-245-4 External links Pappus's hexagon theorem at cut-the-knot Dual to Pappus's hexagon theorem at cut-the-knot Pappus’s Theorem: Nine proofs and three variations Categories: Theorems in projective geometryEuclidean plane geometry

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