Desargues theorem states that if you have two triangles which are perspective to one another then. Nine proofs and three variations x y z a b c a b z y c x b a z x c y fig. Let v be a point and let two triangles be given so that their vertices are distinct from v. Guilhot has formalized in coq a proof of desargues theorem in affine geometry. Kusak has formalized in mizar desargues theorem in. This is an example of a noteworthy theorem of riemannian geometry which has no analogues in more general spaces.
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. Narboux has formalized in coq the area method of chou, gao and zhang 6, 15, 23 and applied it to obtain a proof of desargues theorem in a. Take desarguess theorem of two triangles in distinct planes. Axial perspectivity means that lines ab and ab meet in a point, lines ac and ac meet in a second point, and lines bc and bc meet in a third point, and that these three points all lie on a common line called the axis of perspectivit. Imo training 2010 projective geometry part 2 alexander remorov 1. One needs to understand a few definitions to start with. In projective geometry, desargues s theorem, named after girard desargues, states. Chapter 1 of tondeurs textbook closes with a proof of pappus theorem which we can be stated very compactly, but which we will not prove just now. Girard desargues, the father of projective geometry, proved the following theorem in the 17th century. This illustrates the power and efficiency of our approach using only ranks to prove properties of the projective space.
Since projection preserve incidence, the figures in the plane satisfies desarguess theorem in the plane. Perhaps the most important proposition deduced from the axioms of incidence in projective geometry for the projective space is desargues twotriangle theorem usually stated quite concisely as follows. As emphasized in the literature, the nondegeneracy conditions lead to technical proofs. Chapter 3 solid geometry and desargues theorem math 4520, fall 2017 3. That there is indeed some geometric interest in this sparse setting was first established by desargues and others in their exploration of the principles of perspective art.
Desargues outline finite geometry examples of problems features of desargues another example finite geometry projective geometrya. Later, after we have studied projective geometry, it will serve a very insightful role in tying together several developments of classical euclidean geometry. We say that the two triangle are in perspective from v if the. Projective geometry in projective geometry there are no parallel lines. Two triangles in the real projective plane are in perspective centrally if and only if they are in perspective axially.
Desargues theorem if two triangles are perspective from a point, then they are perspective from a line. Any two lines l, m intersect in at least one point, denoted lm. This article deals with formalizing projective geometry in the coq proof assistant, and studies desargues property both in the plane and in an at least threedimensional setting noted. Triangles d abc and d au bu cu are perspective from a point o if lines aau, bbu and ccu. Chasles et m obius study the most general grenoble universities 3. Drag the points x, y, z, a, a, c, or c to change the triangles.
An application of pappus involution theorem in euclidean and. This theorem is an important milestone toward obtaining the arithmetization of geometry which will allow us to provide a connection between analytic and synthetic geometry. The main theorem of projective geometry that we will use is. Files are available under licenses specified on their description page. Formalizing geometry theorems in a proof assistant like coq is challenging. 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. Kusak has formalized in mizar desargues theorem in the fanoian projective. Click here for a dynamic illustration of desargues theorem geogebra or javasketchpad. Euclidean geometry is mistaken in portraying as coincidental certain results about euclidean points, lines and planes that in fact have a common, uni.
The theorem states that the points, and at the intersections of corresponding sides lie in a line. An application of pappus involution theorem in euclidean. Objects points, lines, planes, etc incidence relation antire. To construct the real projective plane we need to introduce several new points and one new line which contains them all to the euclidean plane. In addition, when considering higherdimensions, the amount of incidence relations e. This is a theorem in projective geometry, more specifically in the augmented or extended euclidean plane.
The work of desargues was ignored until michel chasles chanced upon a handwritten copy in 1845. Desargues theorem is one of the most fundamental and beautiful results in projective geometry. The real projective plane can also be obtained from an algebraic construction. Actually desargues proved the theorem because he was interested in perspective drawings and the geometry it gave rise to is however not projective geometry but what is called descriptive geometry. Pdf a case study in formalizing projective geometry in coq. Nov 29, 20 pappus and desargues finite geometries 1. 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. Meanwhile, jeanvictor poncelet had published the foundational treatise on projective geometry in 1822. Common examples of projections are the shadows cast by opaque objects and motion pictures displayed on a screen. In a projective plane, two triangles are said to be perspective from a point if the three lines joining corresponding vertices of the triangles meet at a common point called the center.
Proving and generalizing desargues twotriangle theorem. His proofs did not use linear algebra which was not developed until the 19th century and are rather more complicated. This approach allows to carry out proofs in a more systematic way and was successfully used to fairly easily formalize desargues theorem in coq. I intend to explain the result, but not all the details. Projective geometry in a plane fundamental concepts undefined concepts. In the plane, proofs are constructed in a traditional way using points and lines. Two triangles are said to be perspective from a line if the three points of intersection of. A quadrangle is a set of four points, no three of which are collinear. This is a partial version of desargues involution theorem see 3, p. Narboux has formalized in coq the area method of chou, gao and zhang, and applied it to obtain a proof of desargues theorem in affine geometry. In plane projective geometry, desargues theorem cannot be proven from the other axioms.
Two triangles are in perspective axially if and only if they are in perspective centrally. Jan 17, 2015 desargues theorem is one of the most fundamental and beautiful results in projective geometry. The usual euclidean plane is contained in what we call the real projective plane. A case study in formalizing projective geometry in coq. Dorrie begins by providing the reader with a short exposition of. A projective point is a line in ir3 that passes through the origin. Denote the three vertices of one triangle by a, b and c, and those of the other by a, b and c. Projective geometry and pappus theorem kelly mckinnie history pappus theorem geometries picturing the projective plane lines in projective geometry back to pappus theorem proof of pappus theorem pappus of alexandria pappus of alexandria was a greek mathematician. In the longer term, the underlying objective of the presented work consists in designing a formal geometry prover able to handle the nondegeneracy conditions, and especially in geometric constraint solving 14, 16. The axiomatic destiny of the theorems of pappus and desargues. Timothy peil, 5 february 20, created with geogebra.
This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. Pdf a case study in formalizing projective geometry in. Research article proving and generalizing desargues two. It is called the desarguesian projective plane because of. Now reverse this process to prove desarguess theorem of. Drag the points l 1 or l 2 of the line to change the axis i. Research article proving and generalizing desargues twotriangle theorem in 3dimensional projective space dimitrioskodokostas department of applied mathematics, national technical university of athens, zografou campus, athens, greece. That is, desargues theorem can be proven from the other axioms only in a projective geometry of more than two dimensions. The dual of this latter characterization permits to state the projective version of menelaus theorem. In the axiomatic development of projective geometry, desargues theorem is often taken as an axiom. Projective geometry, branch of mathematics that deals with the relationships between geometric figures and the images, or mappings, that result from projecting them onto another surface.
All structured data from the file and property namespaces is available under the creative commons cc0 license. The basic notions, and some of the fundamental theorems, of what would later be called projective geometry were first established in the seventeenth century, in response to questions that arose with regard to map projections and problems of perspective encountered by artists in representing threedimensional scenes on planar canvases. So if we prove a theorem for points in a projective plane then the dual result holds automatically for lines. If two triangles are perspective from a line, then they are perspective from a point. Girard desargues 1591 1661, a french architect and mathematician who lived in lyons and paris, was one of the founders of projective geometry. The elegance of their proofs testifies to the power of the method of homogeneous coordinates. In projective geometry, desarguess theorem, named after girard desargues, states. The projective plane is obtained from the euclidean plane by adding the points at infinity and the line at infinity that is formed by all the points at infinity.
I understand this is figure 4 desargues theorem in 3 dimen sions. Dec 05, 2008 a first look at projective geometry, starting with pappus theorem, desargues theorem and a fundamental relation between quadrangles and quadrilaterals. In the riemannian case the only desargues geometries are the euclidean, hyperbolic and elliptic geometries, i. In this geometry, any two lines will meet at one point. Research article proving and generalizing desargues twotriangle theorem in 3dimensional projective space dimitrioskodokostas department of applied mathematics, national technical university of athens, zografou campus, athens, greece correspondence should be addressed to dimitrios kodokostas. We choose to focus on projective geometry which is a simple but powerful. In other words, we can say the triangles are in perspective from the point p. Any two points p, q lie on exactly one line, denoted pq. In this article, we investigate formalizing projective plane geometry.
Desargues theorem and the invariance of the crossratio, were published in a book about perspective by bosse 1648. 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. 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. For readers unfamiliar with projective geometry or unfamiliar with the somewhat dated terminology in dorrie, this one is really hard to read. In two dimensions it begins with the study of configurations of points and lines. 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. Desargues theorem article pdf available in computational geometry 458 october 2012 with 79 reads how we measure reads. Explanation, existence and natural properties in mathematics. Desargues theorem working toward a proof in what follows, we will use four axioms of projective geometry that happen to hold true for rp3. A first look at projective geometry, starting with pappus theorem, desargues theorem and a fundamental relation between quadrangles and quadrilaterals. 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.
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