Please see a chain of six circles associated with a conic. This is a generalization of Pascal theorem, Pappus theorem. I reformulate as following:
Let $1, 2, 3, 4, 5, 6$ be six arbitrary points in a hyperbola. Let $1'$ be arbitrary point in the hyperbola. The circle $(121')$ meets the hyperbola at point $2'$. The circle $(232)$ meets the hyperbola again at $3'$, define points $4', 5', 6'$ similarly. Let circle $(121')$ meets the circle $(454')$ at $A, B$, Let circle $(232')$ meets the circle $(565')$ at $C, D$. Let circle $(343')$ meets the circle $(616') $ at $E, F$. Then six points $A, B, C, D, E, F$ lie on a circle.
1. If $1'$ at $\infty$, six circles are six lines, so the theorem is Pascal theorem.
2. If the hyperbola is two lines, and $1'$ at $\infty$ then six circles are six lines, the theorem is Pappus theorem
My question: Can generalization the result above to Higher Dimensions?