Teorema de pascal y brianchon biography

Then if 2 n of those points lie on a common line, the last point will be on that line, too. If six unordered points are given on a conic section, they can be connected into a hexagon in 60 different ways, resulting in 60 different instances of Pascal's theorem and 60 different Pascal lines. This configuration of 60 lines is called the Hexagrammum Mysticum.

As Thomas Kirkman proved inthese 60 lines can be associated with 60 points in such a way that each point is on three lines and each line contains three points. The 60 points formed in this way are now known as the Kirkman points. There are 20 Cayley lines which consist of a Steiner point and three Kirkman points. Furthermore, the 20 Cayley lines pass four at a time through 15 points known as the Salmon points.

Pascal's original note [ 1 ] has no proof, but there are various modern proofs of the theorem. It is sufficient to prove the theorem when the conic is a circle, because any non-degenerate conic can be reduced to a circle by a projective transformation. This was realised by Pascal, whose first lemma states the theorem for a circle.

His second lemma states that what is true in one plane remains true upon projection to another plane. A short elementary proof of Pascal's theorem in the case of a circle was found by van Yzerenbased on the proof in Guggenheimer This proof proves the theorem for circle and then generalizes it to conics. A short elementary computational proof in the case of the real projective plane was found by Stefanovic We can infer the proof from existence of isogonal conjugate too.

A short proof can be constructed using cross-ratio preservation. Therefore, XYZ are collinear. Another proof for Pascal's theorem for a circle uses Menelaus' theorem repeatedly. Dandelinthe geometer who discovered the celebrated Dandelin spherescame up with a beautiful proof using "3D lifting" technique that is analogous to the 3D proof of Desargues' theorem.

The proof makes use of the property that for every conic section we can find a one-sheet hyperboloid which passes through the conic. There also exists a simple proof for Pascal's theorem for a circle using the law of sines and similarity.

Teorema de pascal y brianchon biography

Pascal's theorem has a short proof using the Cayley—Bacharach theorem that given any 8 points in general position, there is a unique ninth point such that all cubics through the first 8 also pass through the ninth point. In particular, if 2 general cubics intersect in 8 points then any other cubic through the same 8 points meets the ninth point of intersection of the first two cubics.

Pascal's theorem follows by taking the 8 points as the 6 points on the hexagon and two of the points say, M and N in the figure on the would-be Pascal line, and the ninth point as the third point P in the figure. Here the "ninth intersection" P cannot lie on the conic by genericity, and hence it lies on MN. The Cayley—Bacharach theorem is also used to prove that the group operation on cubic elliptic curves is associative.

The same group operation can be applied on a conic if we choose a point E on the conic and a line MP in the plane. Contents move to sidebar hide. Article Talk. Read Edit View history. Tools Tools. Download as PDF Printable version. In other projects. Wikimedia Commons Wikidata item. The 3 long diagonals of a hexagon tangent to a conic section meet in a single point.

Formal statement [ edit ]. Connection to Pascal's theorem [ edit ]. Degenerations [ edit ]. In the affine plane [ edit ]. Proof [ edit ]. See also [ edit ]. References [ edit ]. This result is often called Brianchon's Theorem and it is the result for which he is best known. In fact this theorem is simply the dual of Pascal 's theorem which was proved in :- If all the vertices of a hexagon lie on a conic, and if the opposite sides intersect, then the points of intersection lie on a line.

In [ 1 ] Greitzer points out that Pascal recognised that his theorem was projective in nature so it is surprising that it took years before someone realised that its dual, which is Brianchon's Theorem, would also be true. Brianchon graduated in as first in his class. One might have expected him to continue at this stage to an academic career but these were unusual times in France.

Napoleon Bonaparte had declared an empire in with himself as Emperor. He basically controlled continental Europe and he had only the British to fight against, but without control of the seas he could not mount an invasion. The British under Nelson won a decisive victory at Trafalgar where the Franco-Spanish fleet was destroyed. Napoleon then tried the tactic of blockading Britain and so he gave an order to stop all trade with the British Isles.

However Portugal was reluctant to stop trading with Britain, both for economic and political reasons, and Napoleon decided to send his armies to Portugal to force them to comply with his orders. Although Spain allowed Napoleon's armies to cross their country the campaign was to turn out badly for Napoleon.