Figure 5: Multiple conics passing through four points
[Open a dynamic GeoGebra applet in a new window]
There are various constructions of particular conic sections, such as Euclid's Proposition IV.5, but a geometric construction of an arbitrary conic given five points was first published by William Braikenridge [1733], although Maclaurin disputed his priority in "a rather disagreeable controversy" [Coxeter 1961a, p. 91]. Coxeter gives the construction in both [1961a, p. 91] and [1961b, p. 254]. He suggests that is is based on Pascal's celebrated theorem about the points of intersection of the sides of a hexagon inscribed in a conic section. However, it is not clear that either Maclaurin or Braikenridge knew Pascal's Theorem; see [Mills 1984]. The applet in Figure 6 illustrates that, in general, there exists a conic section passing through any five points.
