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Pascal's Theorem gives a synthetic geometric condition for six points $a,\ldots,f$ in $\mathbb{P}^2$ to lie on a conic. Namely, that the intersection points $\overline{ab}\cap\overline{de}$, $\overline{af}\cap\overline{dc}$, $\overline{ef}\cap\overline{bc}$ are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for $d+4$ points in $\mathbb{P}^d$ to lie on a degree $d$ rational normal curve? In this paper we find many of these conditions by writing in the Grassmann-Cayley algebra the defining equations of the parameter space of $d+4$ ordered points in $\mathbb{P}^d$ that lie on a rational normal curve. These equations were introduced and studied in a previous joint work of the authors with Giansiracusa and Moon. We conclude with an application in the case of seven points on a twisted cubic., 17 pages, 1 figure. Final version. To appear in Bulletin of the London Mathematical Society

We present a generalization of the notion of the orthocenter of a triangle and of Pappus’ theorem. Both subjects were discussed with Pickert in the last year of his life. Furthermore we add a projective Butterfly theorem which covers all known affine cases.

Our point of departure is the following simple common generalisation of the Sylvester-Gallai theorem and the Motzkin-Rabin theorem: Let S be a finite set of points in the plane, with each point coloured red or blue or with both colours. Suppose that for any two distinct points A,B in S sharing a colour there is a third point C in S, of the other colour, collinear with A and B. Then all the points in S are collinear. We define a chromatic geometry to be a simple matroid for which each point is coloured red or blue or with both colours, such that for any two distinct points A,B in S sharing a colour there is a third point C in S, of the other colour, collinear with A and B. This is a common generalisation of proper finite linear spaces and properly two-coloured finite linear spaces, with many known properties of both generalising as well. One such property is Kelly's complex Sylvester-Gallai theorem. We also consider embeddings of chromatic geometries in Desarguesian projective spaces. We prove a lower bound of 51 for the number of points in a 3-dimensional chromatic geometry in projective space over the quaternions. Finally, we suggest an elementary approach to the corollary of an inequality of Hirzebruch used by Kelly in his proof of the complex Sylvester-Gallai theorem., Comment: 23 pages, 3 figures

We give an algorithmic proof for the contrapositive of the following theorem that has recently been proved by the authors:Let S be a finite set of points in the plane, with each point coloured red, blue or with both colours. Suppose that for any two distinct points A and B in S sharing a colour k, there is a third point in S which has (inter alia) the colour different from k and is collinear with A and B. Then all the points in S are collinear.This theorem is a generalization of both the Sylvester-Gallai Theorem and the Motzkin-Rabin Theorem.

In Geometric, right-angled triangle is One Of the two—dimensional plane having three sides With one Of its angle is 9C and whk-h is important to solve problems related to Geometry and sometimes in Other subiect os well. Some fundamental concept 'theorems Of triangles are required to solve such problems and such theorems cre Pythagoras theorern' Pythagoras theorern (ii0 Appollonius theorem Euclids theorem' and (v) Eucli&s 20 theorem (Altitude theorem).3 I n addition to these, the author attempted to develop a new theorem related to right-angled triangle (Maran's theorem of right-angled triangle). The new theorem have been discussed and proved With relevant examples.

The Desargues theorem is a basic theorem in classical projective geometry. In this paper we generalize Desargues theorem in the direction of dynamical systems. Our result comprises an infinite family of configurations, having unbounded complexity. The proof of the result involves constructing special kinds of hyperplane arrangements and then projecting subsets of them into the plane.

A simple geometric proof of Ebisui's theorem, if two triangles A1A2A3 and B1B2B3 are perspective and C3 = A1B2∩A2B1, C1=A2B3∩A3B2, C2=A3B1∩A1B3 then A1A2 A3 and C1C2C3 are also perspective, is given, which is using Desargues's theorem and its converse. With the theorem and an additional theorem, a (154, 203) configuration can be constructed, which is transitive both on points and lines.

In this work is presented a generalization of Poncelet’s theorem to n dimensions which is refered to as the full Poncelet’s theorem. The theorem states that if the reflections of a trajectory by a sequence of confocal quadrics lead to a closed skew polygon, then there exists an (n−1)‐parameter family of polygons having the same property. A physical realization and a projective geometrical proof of this theorem are given. If all the reflecting quadrics coincide, the above theorem reduces to the n‐dimensional Poncelet’s theorem presented by Chang and Friedberg. The geometrical proof is a finite construction based on a preliminary theorem which extends Hart’s lemma. The full Poncelet’s theorem may thus be extended to projective geometries over most fields, including discrete ones.

In the spirit of the Erdos-Szekeres theorem of 1935 we prove some canonical Ramsey Theorems in the plane. In Theorem 1 we are concerned with the number of points in the interior of a convex n -gon. In Theorem 2 we consider a set of points in general position along with a function from the set of points into the plane and prove the existence of a certain canonical configuration. A one-dimensional analog of Theorem 2 is a reformulation of a known theorem concerning intervals on the real line.

For any two lines and a point in R3, there is a line having intersections with these two lines and the point. This fact implies that two lines in R3 make a visual intersection from any viewpoint even if these lines are in twisted position. In this context, the well-known Pappus’ theorem in R2 is simply extended as that in R3, i.e., if the vertices of a spatial hexagon lie alternately on two lines, then from any viewpoint, three visual intersections of opposite sides are visual collinear. In a similar way, Pascal’s theorem is also extended in R3, i.e., if the vertices of a spatial hexagon lie on a cone, three visual intersections of opposite sides are visual collinear from the viewpoint at the vertex of the cone. In this case, for six vertices in R3 we obtain a quartic surface as the set of viewpoints. We will investigate this surface depending on the vertices of a spatial hexagon. A relation between non-singular cubic curve and complete quadrilateral is naturally and geometrically derived.

H.S.M. Coxeter in his book “Introduction to Geometry” quotes a theorem of Mobius. In the paper two counterexamples are given. A corrected version of the theorem is stated and proved.

We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the “projective version” of the well known four vertices theorem for a curve in the Euclidean plane.) We obtain this classical fact as a corollary of some general Sturm-type theorems.

We prove a generalization of the well known Routh's triangle theorem. As a consequence, we get a unification of the theorems of Ceva and Menelaus. A connection to Feynman's triangle is also given.

A polygonal number is a nonnegative integer of the form $$m\frac{{t^{\text{2}} - t}} {2} + t$$ where m is a positive integer, and t is a nonnegative integer. For example, when m = 1, we have the triangular numbers 0, 1, 3, 6, 10, 15, and so on. These are called triangular numbers because n pebbles can be arranged in the form of an isosceles right triangle just in case n has the form (t2 − t)/2 + t.

In a projective plane the set of all points which are on homologous lines of two projective, non-perspective, different pencils with self-homologous common line is called a degenerated Steiner (d. S.) conic. An analytic description for d. S. conics is given. Amitsur's theorem about generalized polynomial identities in division rings finds two geometrical applications: (1) in the discussion of the normal form problem for d. S. conics and (2) in answering the question if a d. S. conic allows beside the defining generation further generations. We study the intersection of d. S. conics with lines and notice that a d. S. conic is never identical with a set of absolute points of a polarity.

(1974). A Theorem about Zig-Zags Between Two Circles. The American Mathematical Monthly: Vol. 81, No. 7, pp. 754-757.

A 4-point Pascal theorem on an oval in a protective plane led to the symmetry theorem in the Minkowski plane, and this relation was used by the author in [2,3] to prove that the symmetry theorem is equivalent to Miquel's theorem and that it implies the tangency theorem (Beruhrsatz). The same special Pascal theorem now leads to an incidence theorem, π, for the Laguerre plane, which is again equivalent to Miquel's theorem. The configuration of π contains 6 points and 5 circles and is thus simpler than that of Miquel, although it does not have the intuitive symmetry properties of the corresponding configuration in the Minkowski plane.

(1980). Quadrangles, Butterflies, Pascal's Hexagon, and Projective Fixed Points. The American Mathematical Monthly: Vol. 87, No. 3, pp. 197-200.

If S is a finite set of points in the plane and no conic contains all points of S, then S determines a conic which contains exactly five points ofS.

We prove duals of Radon's theorem, Helly's theorem, Caratheodory's theorem, and Kirchberger's theorem for arrangements of pseudolines in the real projective plane, which generalize the original versions of those theorems for plane configurations of points. We also prove a topological generalization of the pseudoline-dual of Helly's theorem.

An oval C In a projective plane is a set of triply noncollinear points such that each point of C is on exactly one line which contains no other point of C. An oval has the Pascalian property if each of its hexagons has collinear diagonal points. The author considers those hexagons in which at least two vertices coincide and shows that if every hexagon in this class of 5-point hexagons is Pascalian then the oval has the Pascalian property. Therefore, the 5-point Pascalian property is equivalent to the full Pascalian property. The proof makes use of the coordinatization found in Artzy [1, 2].

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An analogue in a solid of the well known Pascal's theorem (Baker, [1], p. 219) for a conic is established by Baker ([2], pp. 53–54, Ex. 15) after Chasles [6] and by Salmon ([2], p. 142). The same is discussed in detail by Court [8]. The purpose of this paper is to extend it to a projective space of n dimensions or briefly to an n-space Sn. To prove it, we introduce here once again the idea of a set of n+1 associated lines in Sn as indicated in an earlier work (Mandan, [12]) in analogy with a set of 5 associated lines in S4 (Baker, [4], p. 122), and make use of the method of induction.

The circumstances explained in the footnote on p. 61 of the former paper with this title might well have necessitated a re-writing of the whole. Fortunately it appears that only a few short comments are required. For example it may be noted that the first question in §4 can be answered by counting the degrees of freedom in the two configurations. Eight points of a twisted cubic have freedom 20; four pairs of planes drawn at random through four lines of a regulus have freedom 21; therefore the eight planes of the second paragraph of §4 cannot always lead back to the eight points of §2. This is corroborated by the corresponding numbers in [4], which are 31 and 34.

In 1825 the Academie Royale de Bruxelles proposed as a prize topic the extension of Pascal's theorem to space of three dimensions. The prize was won by Dandelin,t who showed that a skew hexagon formed of three lines from each regulus of a hyperboloid of revolution has the Pascal property that pairs of opposite planes meet on a plane, and the dual, or Brianchon property, that the lines joining pairs of opposite vertices meet in a point. Hesset wrote several papers on the IDandelin skew hexagons, emphasizing the polar properties of the Pascal plane and the Brianchon point with respect to the quadric bearing the two reguli. Several of the older analytic geometries of three dimensions devote some space to the problem of the skew hexagon, as Salmon, and more notably Plucker?, who offered much original material. In a recent article, the present writerll has discussed the skew hexagon from an elementary analytic approach. The foregoing citations exhibit the idea that Pascal's theorem deals with six elements of a quadratic curve, and that the space extension offered will deal with six elements of a quadric surface. It happens that an extension of this sort is valid in space of three dimensions; but in S., where n >3, it is clear that a skew hexagon of six rulings of a hyperquadric must lie in an S3 if it is to possess the Brianchon property. It seems, therefore, that the hexagon idea must be abandoned in the search for a valid extension of Pascal's theorem in S, In the plane, a variant of Pascal's theorem affirms that if two triangles are in homology, then the six points of intersection of the sides of the one with the non-corresponding sides of the other lie upon a conic. From a consideration of the converses of this theorem and its dual it occurred in

Publisher Summary This chapter discusses the incidence propositions in the plane. It provides an overview of trivial axioms, duality. A plane projective geometry is an axiomatic theory with the triple 〈Π, Λ, I〉 as its set of fundamental notions and V1, V2, V3 as its axioms, possibly with additional axioms. A hexagon with collinear diagonal points is called a Pascal hexagon. The line on which its diagonal points lie is its Pascal line. The chapter discusses permutation proposition. If in a Pascal hexagon, two adjacent vertices are interchanged and if the resulting figure is a hexagon, then it is a Pascal hexagon. The chapter further discusses weak permutation proposition. If in a Pascal hexagon, two vertices between which lies one other vertex is interchanged and if the resulting figure is a hexagon, then it is a Pascal hexagon. It follows from the permutation proposition that a Pascal hexagon changes into a Pascal hexagon by every permutation of the vertices—provided the result is a hexagon.

Pascal’s classic theorem states that: “the three intersection points of opposite sides of a closed hexagonal line, inscribed in a nondegerated conic, are collinear”. The following extension of Pascal theorem to 3D is considered: “given a closed decagonal line, inscribed in a nondegerated quadric, whose opposite side-lines are secant, the five intersection points of opposite side-lines are coplanary”. (A polygonal line with 10 sides is considered, because 10 — 1 points determine a quadric, as 6 — 1 points determine a conic). Obviously, in this extension to 3D of Pascal theorem some vertices of the polygonal line can not be freely chosen, but an interesting property has been found: the five diagonal lines passing through opposite vertices share a point. This property leads to a simple method to generate the configuration. Moreover, conditions of existence of this configuration are determined and the so called complete configuration is also described in detail. As large expressions appear when coordinates are used, we have developed a package on a computer algebra system that helps us to find and to automatically generate this configuration.

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