The 4th tangent to multiple inconics

A conic is both a second-degree curve, meaning that a straight line intersects it in two points, and a curve of the second class, meaning that from an external point two tangents can be drawn. This exemplifies the duality symmetry between points and lines called duality.

A conic can be considered to be composed of points or to be composed of lines.

A circumscribed conic goes through the three vertices of a triangle; two circumscribed conics go through the vertices and a fourth point. This is the famous fourth intersection of two circumconics.

An inscribed conic is tangent to each of the three sides of the triangle. Two inscribed conics are tangent to these three same lines, and to a fourth line, the famous fourth tangent of two inconics.

At each point on the 4th tangent, the second tangent can be drawn to each conic.

The intersection of lines with conics involves a field extension in the form of an added square root, but the fourth intersection of two circumconics, the first three intersections being at the rational vertices of the triangle, is itself a rational point. Hence this fourth intersection is likely to be a special, central point in the plane of the triangle. The most famous example of this is that the circumcircle and the Steiner ellipse meet at the Steiner point. The dual to this is that the fourth tangent to the Steiner inellipse and the MacBeath inconic is the line that is the dual of the Steiner point, itself an important line in the triangle.

Now it is quite possible for a number of circumconics to share the same fourth intersection, or a number of inconics to share the same fourth tangent. In the case of the circumconics that corresponding points on each conic lie on the a line through the fourth intersection of the two conics. Combined with the idea that a point can be centered, that is, algebraically symmetric to vertices A,B, C, these lines join corresponding special points on each circumconic. This leads to a rather impressive organization of a large number of points in the triangle plane.

The natural map

Let P = ( l : m : n ) be on the line at infinity and let it be central by some reasonable definition. The combination of being on a line and being central is surprisingly restrictive. One can find a reasonable set of points based on simplicity of coordinates. It will turn out that this simple set will generate most of the notable points on lines and conics by what I call the natural mapping.

( : y : ) -> ( : y/M : ) will map a point at infinity to the line Lx + My + Nz = 0.
( : y : ) -> ( : M/y : ) will map a point at infinity to the circumconic L/x + M/y + N/z = 0.
( : y : ) -> ( : y²/M : ) will map a point at infinity to the inconic √(Lx) + √(My) + √(Nz) = 0.

If this natural map projects the same point at infinity to two circumconics, the points on the conics will be collinear with their 4th intersection..

If this natural map projects the same point at infinity to two inconics, the tangents to the points on the conics will concur on their 4th tangent.

Similarly it is possible that a number of inconics share the same fourth tangent. At each point on this fourth tangent a second tangent to each conic can be drawn. If the point on the fourth tangent is a centered point, the second tangent's intersection with the conic will itself be a centered point. Just as the line through corresponding points circumconics will go through their fourth intersection, the tangent at corresponding points on several inconics will concur on their fourth intersection.

The idea of centrality is key. A line has very many points, but not so many centered points. The unifying idea here can be considered to be the centered points on a special line, which I choose to be the line at infinity. Each of those centered points can be mapped to a centered point on a circumconic or inconic. It is precisely these points which are the corresponding points mentioned above. Hence each center on the line at infinity produces a collinear set of points on circumconics sharing the same fourth intersection or concurrent tangents on inconics sharing the same fourth tangent line.

The mapping that moves from the line at infinity to another line or to a conic is a projective map. In coordinates its implementation is in the form of the barycentric product. This product preserves centeredness, very important in the above considerations.

Inconics tangent to ~S

The Lemoine inellipse, the Steiner inellipse, the H-inconic, the Simmons inconics, and the Spieker inconics, and the dual of the circumcircle share ~Sthe dual of the Steiner point as a common fourth tangent. We will show how this organizes points and lines related to these conics.

Figure: The Lemoine and H-inconics are blue, the Simmons conics green, the Steiner inellipse, and the Spieker inconics are red (for the original one) and light red (for the others).

Points and tangents

These are the points involved in the above picture. Most are not in ETC, and in fact, excepting the fact that these points are on the indicated conics, they are not particularly important in other facets of triangle geometry. The relation of each to the other (what I call the natural map) is important.