A Conic Section from Five Elements

 

Fundamental of the construction .

  • Pascal's Hexagon Theorem

    To demonstrate that the three points of intersection of the opposite sides of a hexagon inscribed in a conic section lie on a straight line .
  • Brianchon's Hexagram Theorem

    To demonstrate that the three opposite vertex lines of a hexagram circumscribed about a conic section pass through a point .
  • Desargues' Involution Theorem

    1. The points of intersection of a line with the three pairs of opposite sides of a complete tetragon and a conic section circumscribed about this tetragon form four point pairs of an involution .

    2. The lines joining a point with the three pairs of opposite vertexes of a complete tetragram and the tangents drawn from the point to a conic section inscribed in the tetragram form four ray pairs of an involution .

 

To draw a conic section of which five elements---points and tangents---are known .

I. the five elements are of the same type .

To draw a conic section from five points .

To draw a conic section from five tangents .

II. four elements are of the same type , but the fifth is of the other .

To draw a conic section of which four points 1,2,3,4 and one tangent t are given .

To draw a conic section of which four tangents I,II,III,IV and one point P are given .

 

III. three elements are of one type , two are of the other .

To draw a conic section of which three points A,B,C and two tangents d and e are given .

To draw a conic section of which three tangents a,b,c and two points D and E are given .