CircumconicsTangentGeneration

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.

Generation of circumconics tangent to a line

See Also