In this section, we undertake the study of nondegenerate conics on . Our aim is to relate the algebraic approach of the previous sections to the classical synthetic approach of Chasles and Steiner.

The first important result about conics is that, up to projective transformations, they are all the same. More precisely:

*Remark.*
From now on all conics will be nondegenerate and nonempty.

As a consequence of the previous exercise, we have that it is possible to define the cross-ratio of four points on a conic.

Figure 2.

By duality, we may also define the cross-ratio of four lines tangent to a conic: if , and are four lines tangent to a conic, take a fifth line tangent to the same conic and define the cross-ratio as the cross-ratio of the points , and on the line (Fig. 3)

Figure 3.

Figure 4.

Conversely, we have the following beautiful construction of conics due to Steiner.

is a line passing through

is a conic.

If is a projective transformation taking to and to , then the pencils of lines passing through and are taken to the pencils of vertical and horizontal lines on the plane (Fig. 5). Furthermore, is transformed into the graph of a function from the -axis to the -axis.

Figure 5.

Using Steiner's theorem, we can easily prove that five points determine a conic. Indeed, let , and be five points on the plane. If three or more of these points are collinear then it is easy to see that they all lie on a degenerate conic. Assume now that no three of these points are collinear. Remark that there is a unique projective transformation between the pencil of lines passing through and the pencil of lines passing through such that , , and (Fig. 6). By Steiner's theorem defines a conic that passes through all five points.

Figure 6.

By duality, we have that given five lines, there is a unique conic that is tangent to all of them.

Since five points determine a conic, it is clear that six points are not on the same conic unless they satisfy some special condition. This condition, discovered by the mathematician and philosopher Blaise Pascal, is one of the earliest and prettiest results on projective geometry.

Figure 7.

The following two exercises prove that if the six vertices of a hexagon lie on a conic the three points obtained by intersecting the three pairs of opposite sides are collinear. The proof of the converse uses the same ideas and is left for the reader as a less structured exercise.

Consider the transformation that takes the points of the line to the points of the line that is defined by the following construction:

- If , draw the line .
- Let be the second point of intersection of the line and the conic.
- Draw the line .
- Let be the point .

The dual of Pascal's theorem is the following result originally dicovered by Brianchon.

Figure 8.