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.