A simple proof for the theorems of pascal and pappus marian palej geometry and engineering graphics centre, the silesian technical university of gliwice ul. The dual of pascals theorem is known brianchons theorem, since it was proven by c. As shown in class, there is the rearrangment proof. It states that, given a not necessarily regular, or even convex hexagon inscribed in a conic section, the three pairs of the continuations of opposite sides meet on a straight line, called the pascal line. Projective geometry over f1 and the gaussian binomial coefficients henry cohn 1. Pascal s theorem is a special case of the cayleybacharach theorem. We prove a generalization of both pascals theorem and its converse, the braikenridge maclaurin theorem. The hockey stick theorems in pascal and trinomial triangles. However, a convex hexagon might have a pascal line too far off the diagram to be seen. Looking for patterns solving many realworld problems, including the probability of certain outcomes, involves. W hen we want to use menelaus theorem to prove three certain points lying on a same straight line, we need to specify a triangle such that these three points belong to the three sides of the triangle. We present a proof of poncelets theorem in the real projective plane which relies only on pascals theorem. Pascals theorem article about pascals theorem by the.
Blaise pascal proved that for any hexagon inscribed in any conic section ellipse, parabola, hyperbola the three pairs of opposite sides when extended intersect in points that lie on a straight line. Even as a teenager his father introduced him to meetings for mathematical discussion in paris run by marin. Pascals theorem carl joshua quines from this problem we get our rst two heuristics for pascal s. The theorem of pascal concerning a hexagon inscribed in a conic. Math education geometry pascals mystic hexagram theorem. Pascals theorem is the polar reciprocal and projective dual of brianchons theorem. A bunch of points, all lying on the same circle, with a bunch of intersections is a hint for pascal s, especially if we want to prove a collinearity or concurrence.
There is no field with only one element, yet there is a well defined notion of what projective geometry over such a field means. A pascal theorem applied to minkowski geometry springerlink. The dual to pascal s theorem is the brianchon theorem. Pascals triangle and the binomial theorem mctypascal20091. If a line is drawn from the centre of a circle perpendicular to a chord, then it bisects the chord. A 16 year old discovered this amazing geometry hidden pattern. Brianchon 17831864 in 1806, over a century after the death of blaise pascal. It will take us several easy initial steps, but then the actual proof is very easy.
Projective geometry over f1 and the gaussian binomial. Pascals theorem is a very useful theorem in olympiad geometry to prove the collinearity of three intersections among six points on a circle. The purpose of this article is to discuss some apparently new theorems in projective geometry that are similar in spirit to pascals theorem and brian. Pascals favorite mathematical topic to study, geometry, led to the formulation of pascals theorem. Pascals theorem is a tool for collinearities and concurrences. The special case of a conic degenerating to a pair of lines was known even in antiquity see pappus axiom. Hilbert in, who established that it can be proved for various collections of axioms from the axiom system of euclidean geometry. Addendum to a pascal theorem applied to minkowski geometry. Binomial theorem and pascals triangle 7 excellent examples. If we want to raise a binomial expression to a power higher than 2. It is named after charles julien brianchon 17831864. Its an awesome visual tool and will definitely simplify your work.
Theorem 2 fundamental theorem of symplectic projective geometry. We state a hockey stick theorem in the trinomial triangle too. Pappuss intention was to revive the geometry of the hellenic period 11, p. The big hockey stick theorem is a special case of a general theorem which our goal is to introduce it. Jan 20, 2020 then we will see how the binomial theorem generates pascals triangle. In projective geometry, pascal s theorem formulated by blaise pascal when he was 16 years old determines that a hexagon inscribed in a conic, the lines that contain the opposite sides intersect in collinear points, ie if the six vertices a hexagon are located on a circle and three pairs of opposite sides intersect three intersection points.
From pascals theorem to d constructible curves will traves abstract. If a line is drawn from the centre of a circle to the midpoint of a chord, then the line is perpendicular to the chord. Pascals theorem has an interesting converse, sometimes called the braikenridge maclaurin theorem after the two british. But pascal was also a mathematician of the first order. A simple proof of poncelets theorem on the occasion of its. Survey of geometry table of contents life without geometry is pointless. Pascal published this as essai pour les coniques when he was just sixteen years old. Proof of pascals hexagon theorem 621 if we think of triples representing points and lines as of vectors in the threedimensional space, then the vectors corresponding to a line and a point lying on that line are orthogonal. Pascals theorem article about pascals theorem by the free. There is some intuitive idea of pascals s theorem in. Theorem 1 fundamental theorem of projective geometry.
A bunch of points, all lying on the same circle, with a bunch of intersections is a hint for pascals, especially if we want to prove a. Then we will see how the binomial theorem generates pascals triangle. A very simple proof of pascals hexagon theorem and some. Pascal s theorem is a very useful theorem in olympiad geometry to prove the collinearity of three intersections among six points on a circle.
Prove that the three points of intersection of the opposite sides of a hexagon inscribed in a conic section lie on a straight line. Blaise pascal proved that for any hexagon inscribed in any conic section ellipse, parabola, hyperbola the three pairs of opposite sides when extended intersect in points that lie on a. The dual of pascal s theorem is known brianchons theorem, since it was proven by c. A simple proof for the theorems of pascal and pappus. Pascals hexagram and the geometry of the ricochet con. Online geometry classes, pascal s mystic hexagram theorem proof. Jun 25, 2014 the book the art of the infinite by robert kaplan and ellen kaplan has a wonderful introduction to projective geometry and a proof this this theorem. A 16 year old discovered this amazing geometry hidden. Pascals triangle is an array of numbers, that helps us to quickly find the binomial coefficients that are generated through the process of combinations. If 6 distinct points a, b, c, a, b, and c lie on a conic. In projective geometry, pascals theorem formulated by blaise pascal when he was 16 years old determines that a hexagon inscribed in a conic, the lines that contain the opposite sides intersect in collinear points, ie if the six vertices a hexagon are located on a circle and three pairs of opposite sides intersect three intersection points are colinear. We present a proof of poncelets theorem in the real projective plane which relies only on pascal s theorem. If two sets of k lines meet in k2 distinct points, and if dk of those points lie on an irreducible curve c of degree d, then the remaining k. The book the art of the infinite by robert kaplan and ellen kaplan has a wonderful introduction to projective geometry and a proof this this theorem.
The diagram above shows a very nonconvex hexagon, but since projective geometry does not deal with convexity, a convex hexagon would do just as well. Introduction in 18, while poncelet was in captivity as a war prisoner in. In this last case, the result is usually known as pappuss theorem. A simple proof of poncelets theorem on the occasion of. The dual to pascals theorem is the brianchon theorem.
If on an oval in a projective plane a 4point pascal theorem. A simple proof of poncele ts theorem on the occasion of its bicentennial lorenz halbeisen and norbert hungerbuhler. I n a previous post, we were introduced to pascal s hexagrammum mysticum theorem a magical theorem which states that if we draw a hexagon inscribed in a conic section then the three pairs of opposite sides of the hexagon intersect at three points which lie on a straight line. Pascal s triangle and the binomial theorem task cardsstudents will practice finding terms within pascal s triangle and using pascal s triangle and the binomial theorem to expand binomials and find certain terms. A simple proof of poncelets theo rem on the occasion of its bicentennial lorenz halbeisen and norbert hungerbuhler. At the age of sixteen, he wrote a significant treatise on the subject of projective geometry, known as pascals theorem, which states that, if a hexagon is inscribed in a circle, then the three intersection points of opposite sides lie on a single line, called the pascal line. Pascals theorem is a special case of the cayleybacharach theorem. In projective geometry, pascals theorem also known as the hexagrammum mysticum theorem states that if six arbitrary points are chosen on a conic i. Generalizing the converse to pascals theorem via hyperplane. The important role of pascal s proposition in the construction of geometric systems over an infinite field was first investigated by d. Today, we will use menelaus theorem to prove pascals theorem for the circle case. Mathematical induction, combinations, the binomial theorem and fermats theorem david pengelleyy introduction blaise pascal 16231662 was born in clermontferrand in central france. High school, honors geometry, college, mathematics education.
Looking for patterns solving many realworld problems, including the probability of certain outcomes, involves raising binomials to integer exponents. Jul 31, 2009 pascal s theorem in projective geometry. The common pascal is the polar of qwith respect to the conic. Pascals theorem belongs to projective geometry it really has nothing to do with lengths or angles, and this is why it is initially hard to see how we might prove it. In geometry, brianchons theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals those connecting opposite vertices meet in a single point. A bunch of points, all lying on the same circle, with a bunch of intersections is a hint for pascals, especially if we want to prove a collinearity or concurrence. David pengelley introduction blaise pascal 16231662 was born in clermontferrand in central france. Pascals theorem carl joshua quines from this problem we get our rst two heuristics for pascals. Proof of pascals hexagon theorem 621 if we think of triples representing points and lines as of vectors in the threedimensional space, then the vectors corresponding to a.
Pascal s theorem is the polar reciprocal and projective dual of brianchons theorem. Pascals triangle and the binomial theorem task cardsstudents will practice finding terms within pascals triangle and using pascals triangle and the binomial theorem to expand binomials and find certain terms. His great work a mathematical collection is an important source of information about ancient greek mathematics. References 1 artzy, r a pascal theorem applied to minkowski geometry. This notion is familiar to experts and plays an interesting role behind the scenes in combinatorics. Pascal s favorite mathematical topic to study, geometry, led to the formulation of pascals th eorem. It was formulated by blaise pascal in a note written in 1639 when he was 16 years old and published the following year as a broadside titled essay povr les coniqves. The important role of pascals proposition in the construction of geometric systems over an infinite field was first investigated by d.
Survey of geometry minnesota state university moorhead. Generalizing pascals mystic hexagon theorem will traves department of mathematics. If one is given six points on a conic section and makes a hexagon out of them in an arbitrary order, then the points of intersection of opposite sides of this hexagon will all lie on a single line. Pascal s favorite mathematical topic to study, geometry, led to the formulation of pascal s theorem. This states that pairs of opposite sides of a hexagon inscribed in any conic section meet in three collinear points. Theorem 2 is false for g 1 since in that case t p2gk is a discrete poset. Nine proofs and three variations x y z a b c a b z y c x b a z x c y fig. Note that pascals theorem is true regardless of where the points lie on the conic. Blaise pascal 16231662 french mathematician, philosopher and inventor, discovered his famous theorem at the age of 16, in 1640, and produced a treatise on conic sections entitled essai pour les coniques. There are many different ways to prove this theorem, but an easy way is to use menelaus theorem. Mathematical induction, combinations, the binomial theorem and fermats theorem. Pascals theorem has an interesting converse, sometimes called the braikenridge maclaurin theorem after the two british mathematicians william braikenridge and colin maclaurin.
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