Construction of Conics from Pascal's Theorem

Given five points on a conic, Pascal's theorem helps find arbitrarily many points on the same conic. Let, for example, points A, B, C, D, E are known to be located on a conic. If F is a sixth point on the conic then, according to Pascal's theorem, the intersections P = AD ∩ BC, Q = AF ∩ BE, and R = CF ∩ DE are collinear.

The converse of Pascal's theorem, proved independently by W. Braikenridge and C. MacLaurin [Coxeter & Greitzer, p. 76] states that

If F is not given and X is a random point, then define Q = PX ∩ BE, and R = PX ∩ DE and subsequently F = AQ ∩ CR. By Pascal's theorem F is guaranteed to lie on the conic. Thus, given A, B, C, D, E, any other point on the conic can be constructed this way by varying X over the plane.

The applet below illustrates this construction.

The derivative of this construction is the method of MacLaurin. To understand how it works, observe what happens when you drag point X. Point P remain fixed, Q slides along CD while R slides along BC. None of other points moves, except for F, which remains on the conic as expected.

The formulation of MacLaurin's construction begins with a variable triangle QRF whose side lines QR, FQ, FR pass through the fixed points P, A, E, while the vertices Q and R move along the lines CD and BC. The construction also fixes B and D - the intersection of AP with the second line and that of EP with the first. Under these constraints vertex F describes the conic that passes through A and E and the intersection C of the two given straight lines.

References

1. H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, MAA, 1967

Conics

• Conic Sections
• Conic Sections as Loci of Points
• Construction of Conics from Pascal's Theorem
• Cut the Cone
• Dynamic construction of ellipse and other curves
• Joachimsthal's Notations
• MacLaurin's Construction of Conics
• Newton's Construction of Conics


• Ellipse
• Parabola
  

  Pascal and Brianchon Theorems

  • Pascal's Theorem
  • Pascal in Ellipse
  • Pascal's Theorem, Homogeneous Coordinates
  • Projective Proof of Pascal's Theorem
  • Pascal Lines: Steiner and Kirkman Theorems
  • Brianchon's theorem
  • Brianchon in Ellipse
  • The Mirror Property of Altitudes via Pascal's Hexagram
  • Pappus' Theorem
  • Pencils of Cubics
  • Three Tangents, Three Chords in Ellipse
  • MacLaurin's Construction of Conics
  • Pascal in a Cyclic Quadrilateral
  • Parallel Chords
  • Parallel Chords in Ellipse
  • Construction of Conics from Pascal's Theorem
  • Pascal: Necessary and Sufficient
  • Diameters and Chords
  • Chasing Angles in Pascal's Hexagon
  

  

  Copyright © 1996-2009 Alexander Bogomolny