Brianchon's theorem apply on degenerate conic

Question

Brianchon's theorem states that when a hexagon is circumscribed around a conic section (a hexagon formed by six tangent lines of a conic section), its principal diagonals (those connecting opposite vertices) meet in a single point.

I'm thinking if this theorem can apply on a degenerate conic, e.g. 2 intersecting or parallel lines. So there is a weird question: how to define 6 tangent lines of 2 intersecting or parallel lines?

If we apply Pascal's theorem on degenerate conic, we get Pappus's theorem. Since Brianchon's theorem is the dual of Pascal's theorem, Brianchon's theorem ondegenerate conic should be the dual of Pappus's theorem mentioned here with below diagram:

Is there any relation between above diagram and 6 tangent lines of 2 intersecting or parallel lines? (Obviously 6 tangent lines are ABCabc, but where are tangent points and 2 intersecting or parallel lines?)

Answer 1

For a degenerate conic that is two intersecting lines, the tangents are the pencil of lines going through the intersection. So the only Brianchon hexagon will have vertices that are identical.

The diagram you present can be thought of as a Brianchon hexagon for the degenerate conic that is the two points $G$ and $H$. It is a line conic, i.e. a conic that is defined as the envelope of a set of tangents. The tangents in this case are the two pencils of lines going through $G$ and $H$. And $G$ and $H$ can be thought of as the points that are dual to the outer straight lines in Pappus' Theorem. The mapping of a point conic to a line conic is an essential part of the dualization of Pappus' Theorem.

The figure above is from Richter-Gilbert, Perspectives on Projective Geometry, pg 161 and shows a progression of line conics going from ellipse to hyperbola, going through the two point line conic.

Answer 2

brainjam's answer already covers the core aspect. This here is an extension to that. I'll focus on some aspects of duality.

Projective duality exchanges the concepts "point" and "line". With that you'd also exchange "collinear" with "concurrent", "join" with "meet" and so on. Doing all of this for Pappus' configuration, you end up with that very same configuration. In that configuration, any single incidence can be considered the result of all the others. So Pappus' theorem is its own dual.

The dual of a conic is again a conic. The dual of defining a conic as a set of incident points is defining a conic as a set of tangent lines. The dual to a degenerate conic that has all its incident points lie on a pair of lines is a degenerate conic that has all its tangent lines pass through a pair of points. The former would have all its tangent lines pass through the point of intersection, while the latter has all its incident points on the line connecting the two points. So when your question asks about a pair of distinct lines making up the conic, that's not applicable: the two lines of the degenerate conic in primal view coincide and since a line connecting two points isn't adding information to the configuration, you usually wouldn't even include it in the figure.

Algebraically you'd write $\{p\mid p^T\cdot A\cdot p=0\}$ for the set of points $p$ on the conic and $\{l\mid l^T\cdot B\cdot l=0\}$ for the set of tangent lines $l$. Both of these are expressed as $3\times3$ matrices where $A\cdot B=\lambda I$ i.e. the product of the matrices is some multiple of the unit matrix. For non-degenerate conics, the matrices have full rank and you can compute one from the other using matrix inversion. In the case of a pair of lines, $A$ has rank $2$ and you can still compute $B$ as it's adjunct, resulting in a rank one matrix describing the point of intersection. For a conic defined as a pair of points, $B$ has rank $2$ and $A$ has rank $1$, i.e. a single line of algebraic multiplicity two. So you can describe that kind of conic as a pair of matrices, or only as the matrix used for the set of tangent lines. A description as a set of points no longer carries enough information to fully describe that kind of conic, which in my opinion is the reason why people are less familiar with this: usually we learn about conics as sets of points.