Construction of Conics from Pascal's TheoremGiven 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 F is not given and X is a random point, then define 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 Conics
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