(Note: I'm not a native English speaker.)
When I was playing around with Geogebra, I personally discovered some interesting properties concerning conic sections. Here is one of them.
Consider the following configuration:
Draw three conics $C_1$ $C_2$ $C_3$ such that each pair of conics has two intersection points. Denote one of the intersetion points of $C_1$ and $C_2$ as $A_1$. Similarly, denote one of the intersetion points of $C_2$ and $C_3$ as $A_2$, and one of the common points of $C_3$ and $C_1$ as $A_3$. Take an arbitrary point $P_0$ on $C_1$. Draw line $P_0A_1$ and denote the second intersection point of $C_2$ and $P_0A_1$ as $P_1$. Draw line $P_1A_2$ and denote the second intersection point of $C_3$ and $P_1A_2$ as $P_2$. Draw line $P_2A_3$ and denote the second intersection point of $C_1$ and $P_2A_3$ as $P_3$. This is the first iteration, and you can define $P_4, P_5 , P_6, \ldots$ by continuing this process: let $P_3A_1$ meet $C_2$ again in $P_4$, and so on.
Normally $P_3, P_6, P_9, \ldots$ don't concur with $P_0$. However, if you change the position, shape and size of $C_1$ properly, you will get a configuration where $P_9$ always coincides with $P_0$, no matter where $P_0$ is. (or always $P_{6}=P_0$, $P_{12}=P_0$, etc.)
Here is an example. In this figure, The line $P_8A_3$ always passes through $P_0$, meaning that $P_9$ always coincides with $P_0$, and $P_0P_1P_2P_3P_4P_5P_6P_7P_8$ forms a closed 9-gon.
Figure 1a
Figure 1b
It seems that the same thing still holds even if you add more conics $C_4, C_5, \ldots$.
Figure 2a
Figure 2b
Then I also found that the same thing still happened when I added arbitrarily taken fixed points $B_1,B_2,B_3$ (see the figures below). Again, you need to adjust the position and size of $C_1,C_2,C_3$ properly, but it is not that difficult to get a configuration on Geogebra such that $P_{12}$ always coincides with $P_0$.
Figure 3a
Figure 3b
(Follow the path like $P_0→A_1→P_1→B_1→P_2→A_2→P_3→B_2→P_4→A_3→ \ldots$)
A special case: $n$ intersecting circles chainwise and $2n$ intersection points ($n\geq3$). In this case, no matter how you draw $n$ circles, it holds that $P_{2n}=P_0$.
Figure 4
More interesting properties can be observed for $n=3$, just see the figure below for details.
Figure 5
So what is this? Is this a direct consequence of Poncelet's porism or something related to that? What is this theorem called?