## Generation of circumconics tangent to a line

Consider a triangle ABC and a line L. The set of circumconics of ABC which are tangent to line L has a nice structure which was discussed in CircumconicsTangentToLine.html . Here is a way to generate such a conic through line intersections.
Let the conic c pass through the vertices of ABC and be tangent to line L at its point Q. Take an arbitrary point R on line L. Join R and Q with the vertices of the precevian triangle A0B0C0 of ABC with respect to the line L. Through the intersection points of the created lines result quadrangles. Among them there are three having the sides of A0B0C0 as diagonals. The other diagonals of these three quadrangles pass through the same point P lying on c.

The property is a result of an analogous property for parabolas studied in CircumparabolaGeneration.html .
In fact, consider the projectivity (f) which fixes the vertices of ABC and maps the tripole T of L with respect to triangle ABC to the centroid of ABC. Then L maps via (f) to the line at infinity and the property reduces to the corresponding for parabolas discussed in the aforementioned reference.