Actually only half of it is called desargues theorem perspectivity from a point implies perspectivity from a line whereas the other half is called converse of desargues theorem. Similarly, euclidean space can be regarded as the set of ordered triples of real numbers. The two given green and red triangles are said to be in a perspective position, that is, the lines through corresponding pairs of vertices are concurrent. The desargues configuration when desargues theorem holds in a projective plane we get ten points and ten lines with each line containing exactly three of the ten points and any three lines intersecting at exactly one of the ten points. Explanation, existence and natural properties in mathematics. Theorem 2 pappus involution theorem the three pairs of opposite sides of a complete quadrangle meet any line not through a vertex in three pairs of an involution. Desarguess theorem is therefore one of the simplest geometric theorems whose natural home is in. The axiomatic destiny of the theorems of pappus and desargues. Pdf a case study in formalizing projective geometry in coq. If the plane of i is parallel to the plane of i, r is also a circle.
An analytic proof of the theorems of pappus and desargues. Desargues theorem article pdf available in computational geometry 458 october 2012 with 79 reads how we measure reads. Introduction mathematical and statistical sciences. Im not perfectly sure, but i assume that pascals theorem should be not only a generalization of but also a consequence of pappos theorem, which in turn means you have a projective plane over some field. Apr 10, 2012 projective geometry class modeling desargues theorem. In this article we will use the desargues theorem and its reciprocal to solve two problems. Girard desargues 21 february 1591 september 1661 was a french mathematician, architect and engineer, who is considered one of the founders of projective geometry. A case study in formalizing projective geometry in coq. Pappus and desargues finite geometries linkedin slideshare. An application of pappus involution theorem in euclidean and noneuclidean geometry.
Nov 29, 20 pappus and desargues finite geometries 1. Dual of desargues theorem university of south carolina. His work was rediscovered at the beginning of the 19th century, and one of his results became known as desargues. This proof of desargues theorem shows us how to draw the. Since we have not listed the axioms for a projective geometry in 3space, we will not discuss the proof of the theorem here, but the proof.
This is a theorem in projective geometry, more specifically in the augmented or extended euclidean plane. For cevas theorem you obviously need lengths, unless you. Triangles d abc and d au bu cu are perspective from a point o if lines aau, bbu and ccu meet at o. A projective involution is a projective transformation of period two. Desargues theorem and the invariance of the crossratio, were published in a book about perspective by bosse 1648. The theorem states that if two triangles abc and a. Kevin casto and desislava nikolov converse desargues theorem. We can regard the euclidean plane as defined as the set of ordered pairs of real numbers. On menelaus theorem singapore mathematical society. Proving and generalizing desargues twotriangle theorem. The point is distinct from each vertex of at least one of the simplexes. In order to make the statement of the theorem of desargues easier, we make.
Desargues 15911661 two triangles are perspective relative to a point, if and only if they are perspective relative to a line. Objects points, lines, planes, etc incidence relation antire. Triangles d abc and d au bu cu are perspective from a line l if the points x bc 9 bu cu,y ac 9 au cu. Media in category desargues theorem the following 19 files are in this category, out of 19 total. Desargues theorem, the desargues graph, and the desargues crater are named in his honour. The beach towel theorem given a beach towel on a level beach, a second beach towel of the same size and shape, oriented in the same direction, is completely determined by the closest corner. Pappus implies desargues we assume that pappus theorem holds and show that desargues theorem also holds. As well as presenting some olympiad problems which can be solved with this theorem. Assignment construct a triangle abc and choose a point p outside the triangle. The videos argument is not rigorous because we have not yet explained the axioms behind projective geometry. Nine proofs and three variations x y z a b c a b z y c x b a z x c y fig. I intend to explain the result, but not all the details.
We prove that the well known ceva and menelaus theorems are both particular cases of a single theorem of projective geometry. Given a beach towel on a level beach, a second beach towel of the same size and shape, oriented in the same direction. Pappus theorem if points a,b and c are on one line and a, b and c are on another line then the points of intersection of the lines ab and ba, ac and ca, and bc and cb lie on a common line called the pappus line of the configuration. That means, the theorem remains true if points and lines are interchanged. Pdf two applications of desargues theorem florentin. Trilateral a, b, c with dotted lines and trilateral a, b and c are in perspective from the line v the emboldened line. If we draw this standard oil derrick like picture for desargues theorem, we can read the statment and conclusion from the picture. Two triangles that are perspective from a point are perspective from a line, and converseley, two triangles that are perspective from a line are perspective from a point. Now reverse this process to prove desargues s theorem of two triangles in the plane. For that, abc and def are each two round triangles with dual triangles abc and def, respectively, and p a point on each of the circles adad, bebe, and cfcf. Two triangles are perspective from a point if and only if they are perspectivefrom. It is selfdual in the sense that the following exchanges. In their most basic form, cevas theorem and menelauss theorem are simple formulas of triangle geometry.
Desarguess theorem, in geometry, mathematical statement discovered by the french mathematician girard desargues in 1639 that motivated the development, in the first quarter of the 19th century, of projective geometry by another french mathematician, jeanvictor poncelet. Explanation, existence and natural properties in mathematics a case study. Desarguess theorem, as illustrated by eves 9, veblen and young 28, and horadam 16. Thus there is one series of parallel planes which intersect the cone in circles. In other words, we can say the triangles are in perspective from the point p. Solid geometry and desargues theorem math 4520, fall 2017 3. An application of pappus involution theorem in euclidean. Projective geometry class modeling desargues theorem.
Research article proving and generalizing desargues two. Dorrie begins by providing the reader with a short exposition of. For beginning we will enunciate and prove desargues theorem. The theorem of desargues is true in the real projective plane, but not in every projective plane. In this article w egiv an analytic proofpappus theorem and. Desargues s work was well known by his contemporaries, but half a century after his death he was forgotten. Desargues brouillon project and the conics of apollonius. Nicolas magaud, julien narboux, pascal schreck universit. The theorem states that the points, and at the intersections of corresponding sides lie in a line.
Readers are expected to be familiar with projective. Recall that all lines extend to infinity in both directions, even if we draw only some segments on them. For the proof of the theorem two cases are to be considered. Now reverse this process to prove desargues s theorem of two triangles in. An application of pappus involution theorem in euclidean and. Since projection preserve incidence, the figures in the plane satisfies desargues s theorem in the plane. Coxeters diagram of desarguess theorem, with crannells labels. In some countries this may not be legally possible. Monges theorem and desargues theorem, identified monges theorem says that for three disjoint circles of unequal radii, with no one contained in any other, the pairs of external tangents meet in three points that are collinear. That is, desargues theorem can be proven from the other axioms only in a projective geometry of more than two dimensions.
Before we state the theorem and investigate a problem or two using desargues theroem, it might be helpful to understand what exactly it means. Take desargues s theorem of two triangles in distinct planes. E and f are all both in the plane of the red triangle, abc and in the plane of the green triangle, a 0. Pdf in this article we will use the desargues theorem and its reciprocal to solve two problems. The topics include desarguess theorem, harmonic conjugates, projectivities, involutions, conics, pascals theorem, poles and polars. For n 3, the above stated theorem is obviously the theorem of desargues. A polychromatic proof of desargues theorem 455 desargues theorem turns entirely on the intersections of the ariousv planes. One needs to understand a few definitions to start with. Let v be a point and let two triangles be given so that their vertices are distinct from v. In that case da becomes a tangent to the conic and the theorem of desargues beomes a property of triangles inscribed in conics. Desargues theorem by mark freitag one of the most fundamental theorems in projective geometry is desargues theorem.
If desargues, the daring pioneer of the seventeenth century, could have foreseen what his ingenious method of projection was to lead. The original version of desargues theorem appeared in a. Consider the intersection r of a plane not through a and the conic surface. This work has been released into the public domain by its author, i, jtico. Since we have not listed the axioms for a projective geometry in 3space, we will not discuss the proof of the theorem here, but the proof is similar to the argument made in the illustration above. Ion patrascu, the national college fratii buzesti, craiova, romania in this article we will use the desargues theorem and its reciprocal to solve two problems. For readers unfamiliar with projective geometry or unfamiliar with the somewhat dated terminology in dorrie, this one is really hard to read. Choose points aon the line pa, bon pb, and con pc and construct the triangle abc.
Desargues never published this theorem, but it appeared in an appendix entitled universal method of m. The foregoing drawing exercise illustrates a property of desargues theorem called selfduality. In projective geometry, desarguess theorem, named after girard desargues, states. Pdf two applications of desargues theorem researchgate. The theorem of desargues transfers with the same wording to conics defined in the complex projective space. Desargues outline finite geometry examples of problems features of desargues another example finite geometry projective geometrya. This is a partial version of desargues involution theorem.
To see a representation of this theorem use the interactive sketchpad. In the axiomatic development of projective geometry, desargues theorem is often taken as an axiom. Desargues brouillon project and the conics of apolloniirs a 5 figure i. To prove this assume that the two triangles abc and a0b0c0are perspective relative to a point p and apply three times the theorem of menelaus menelaus theorem. Generalized theorems of desargues for wdimensional. We prove configuration theorems that generalize the desargues, pascal, and pappus theo rems. Desargues theorem, the desargues graph, and the crater desargues on the moon are named in his honour. On the desargues involution theorem markbcc168 september 8, 2017 as the title suggests, this article will deal with one powerful theorem in projective geometry, desargues involution theorem and its variants. The main theorem of projective geometry that we will use is. You have constructed two triangles which are perspective with respect to p.
The topics include desargues s theorem, harmonic conjugates, projectivities, involutions, conics, pascals theorem, poles and polars. Desargues theorem working toward a proof in what follows, we will use four axioms of projective geometry that happen to hold true for rp3. Because padbcbd 11 1 1 it results that the triangles dc d11. Girard desargues, born february 21, 1591, lyon, francedied october 1661, france, french mathematician who figures prominently in the history of projective geometry. Converse desargues theorem kevin casto and desislava nikolov brown university university of the free state october 1, 2012. Three or more line segments in the plane are concurrent if they have a common point of intersection. Actually only half of it is called desargues e orem perspectivity from a point implies perspectivity from a line whereas the other half is called converse of desargues eo rem. Chapter 2 a polychromatic proof of desargues theorem. Choose points a on the line pa, b on pb, and c on pc and construct the triangle abc. A special case arises when d tends to coincide with a. Applying the reciprocal theorem of desargues it results that these triangles are homological, that is, the lines. Florentin smarandache, university of new mexico, u. I, jtico grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law. Aa1 and cb1 of triangles aad11 and cbc11 intersect in the collinear points p,bd.
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