### Construction of Conics

**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.

**Figure 6: A conic section passing through five points**

[Open a dynamic GeoGebra applet in a new window]