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 = AB ∩ DE, Q = CD ∩ AF, and R = BC ∩ EF are collinear.

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

If the three pairs of opposite sides of a hexagon meet at three collinear points, then the six vertices lie on a conic, which may degenerate into a pair of lines.

If F is not given and X is a random point in the plane, then define Q = PX ∩ CD, and R = PX ∩ BC and subsequently F = AQ ∩ ER. 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.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

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
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