Poncelet's closure theorem is an important theorem in mathematics, which reveals the relationship among conic curves such as ellipse, parabola and hyperbola. The expression of this theorem is somewhat abstract, but it can be understood through some examples. For example, for pentagonal ABCDE, if it is inscribed on ellipse O 1 and circumscribed on ellipse O2, then any point A' on ellipse O 1 is tangent to ellipse O 1 in b', then it is tangent to ellipse O 1 in c', so it is C.

Penseri's closure theorem can also be used to prove the group structure relationship between conic curves. For example, for two conic curves $Lambda$ and $Gamma$, if they intersect at four points (the tangent point is calculated repeatedly), then these four points determine a conic system $Omega$. Consider n-sided $PQR$ inscribed in $Lambda$ and circumscribed by $Omega$ and n conic curves $ gamma _ i $,$ i = 1, 2,3, cdots and n $. After proving the general situation of n by mathematical induction, let $ gamma _ 1 =