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
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
The applet below illustrates this construction.
|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.
- H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, MAA, 1967