Poncelet's Theorem in the four nonisomorphic finite projective planes of order 9

Abstract

We study Poncelet's Theorem in the four nonisomorphic planes of order 9. We first prove that Poncelet's Closure Theorem holds for the coordinate plane over the Galois field with 9 elements. On the contrary, we will see that the other three nonisomorphic planes of order 9, which are constructed over the miniquaternion near-field of order 9, will fail to be Poncelet planes. This gives a complete discussion of Poncelet's Theorem in finite projective planes of order 9. In addition, we offer a proof of Poncelet's Theorem for triangles in all finite projective coordinate planes which is based upon Pascal's Theorem.