# Family of conics generated by two conics

Let $$a,b$$ be conics as shown in the graph below, let $$K$$ be a point on $$a$$ and draw the two tangents of $$b$$ from it, intersecting $$a$$ at $$L,M$$ respectively. Then, draw tangents of $$b$$ from $$L,M$$, intersecting $$a$$ at $$O,R$$. This process can be iterated until the two intersection coincide. If this happens, the Poncelet's porism states that the coincidence relation remains as $$K$$ varies on $$a$$. Else, we obtain an infinite collection of lines.

Take every intersection of the lines in the collection. As $$K$$ varies on $$a$$, I observe that the locus of every intersection seem to be some conic, and many of them share the same one. This seems to generate an infinite(?) family of conics from the two original ones, each contains infinite(?) points that belong to the intersection of the line collection above. I made serveral attempts trying to proof the statement, but got no positive results. The points whose locus are the same conic do not show explicit pattern. I believe that if the points are indeed on some conic, the iterate process showed above may represent some projective transform so every two lines in the collection are projectively correspond. However, I have no knowledge dealing with projectives from one conic to another.

Question: Is the conjuecture above true? Are there any underlying explanation of this fact?

Any help would be appreciated.

• That several points lis on the same locus is not a surprise. Take for instance $P$, intersection of the two tangents from $K$ and $M$. When $K\to M$ then $P\to W$, hence it is obvious that $W$ is on the locus of $P$. You must only prove that the locus of any intersection is a conic. May 24 at 17:57

The phenomenon you're seeing is related to Poncelet's General Theorem and is explained by a theorem of Darboux that is the (projective) dual of Poncelet's and is described in Section 9 of Del Centina, Poncelet’s porism: a long story of renewed discoveries, I.

You can play with a related construction, which is to consider the line $$LM$$. As $$K$$ varies, you'll find that $$LM$$ is tangent to a conic. You can do that for any line connecting points that you have constructed on $$a$$, and you'll find that these lines as well will envelope conics, due to Poncelet's General Theorem. This family of conics will belong to the conic pencil generated by $$a$$ and $$b$$.

$$a$$ and $$b$$ also define a so-called tangential pencil of conics, and by Darboux's Theorem, the loci of $$P$$ and the other intersections are conics that are members of this tangential pencil.

My answers to two other questions on math.SE give more detail and references. I suggest you follow up with them, because they give information that I'd otherwise have to duplicate here. Note that Poncelet's theorem applies also to pencils of circles (coaxal circles) and much of the literature focuses on that case.

For Poncelet's theorem, here are the question and my answer.

For Darboux's Theorem, here are the question and my answer.

For both questions there are answers other than my own, and they may suit your purposes more than mine, so look around. Same for the comments.