Conic Sections

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## Hexagrammum MysticumLet's conclude with a final synthetic result, which we state rather than prove. Everyone knows that two points determine a line. Again, every three noncollinear points determine a circle, the
The first true theorem of projective geometry was discovered by Pappus of Alexandria (c.290 – c.350): if the six vertices of a hexagon lie on two lines in alternation, then the three points of intersection for pairs of opposite sides are collinear. Here a "hexagon" is taken as the figure formed by joining six vertices in order, whether or not the "sides" intersect. Notice that line segments have no significance in projective geometry, as we need a concept of betweenness to define them; the sides of the hexagon are lines. Interest in projective geometry was renewed during the Renaissance through its applications in perspective drawing and architecture. Great artists such as Leonardo da Vinci (1452 – 1519) and Albrecht Dürer (1471 – 1528) wrote books on the mathematics of perspective; the architect Girard Desargues (1591 – 1661) studied nonmetrical geometry and conic sections as well, and has been called the father of projective geometry. In projective geometry, there is only one type of conic. The distinction between the hyperbola, the parabola, and the ellipse belongs to geometry as we usually conceive of it. We have seen that three noncollinear points determine a circle; similarly, five points in This was made concrete by the great mathematician, scientist, and philosopher Blaise Pascal (1623 – 1662) when he was just sixteen. He proved that if a hexagon is inscribed in a conic, then the three points of intersection for pairs of opposite sides are collinear. This theorem is known as the A century and a half later, this theorem was "dualized" by the French mathematician Charles Julien Brianchon (1783 – 1864), who proved that, if a hexagon is circumscribed about a conic (with its sides tangent to the conic), its three diagonals are concurrent (meet in a single point). His paper can also be found in ## EpilogueThis concludes our tour through the theory of conic sections. I hope you've enjoyed it. The material formed the basis for a Topics in Mathematics course taught at Rio Grande College in 2012. We began with an overview of Euclidean geometry, emphasizing similarity and Apollonian circles. The main discussion of conics concluded with the derivation of their familiar representations in Cartesian coordinate systems; we then moved on to quadric surfaces. These pages are humbly dedicated to Robert Miller, my algebra and calculus teacher at Medina Valley High School. I mentioned him in our discussion of satellite dishes. I was a C student in math at the time and had my share of social problems, and if he hadn't taken an interest in me for some inexplicable reason then I would never have dreamed of pursuing math beyond high school. He affected my life in a profound way, though I don't suppose he ever knew it. |

**Michael Ortiz, Ph.D.
Assistant Professor of Mathematics
Department of Natural and Behavioral Sciences**

Sul Ross State University Rio Grande College

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