other point (c,d′) with the same first coordinate. Any involution of an elliptic curve with a fixed point, when expressed in the group law, has the form x → p − x for some p, so has this form. Similarly, the projection X → D is a degree 2 morphism ramified over the contact points on D of the four lines tangent to both C and D, and the corresponding involution has the form x → q − x for some q. Thus the composition is a translation on X. If a power of has a fixed point, that power must be the identity. Translated back into the language of C and D, this means that if one point c ∈ C (equipped with a corresponding d) gives rise to an orbit that closes up (i.e., gives an n-gon), then so does every point. The degenerate cases in which C and D are not transverse follow from a limit argument. See also Finding Ellipses Hartshorne ellipse Steiner's porism Tangent lines to circles References Bos, H. J. M.; Kers, C.; Oort, F.; Raven, D. W. "Poncelet's closure theorem". Expositiones Mathematicae 5 (1987), no. 4, 289–364. External links David Speyer on Poncelet's Porism D. Fuchs, S. Tabachnikov, Mathematical Omnibus: Thirty Lectures on Classic Mathematics Interactive applet by Michael Borcherds showing the cases n = 3, 4, 5, 6, 7, 8 (including the convex cases for n = 7, 8) made using GeoGebra. Interactive applet by Michael Borcherds showing Poncelet's Porism for a general Ellipse and a Parabola made using GeoGebra. Interactive applet by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 3) made using GeoGebra. Interactive applet by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 5) made using GeoGebra. Interactive applet by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 6) made using GeoGebra. Java applet showing the exterior case for n = 3 at National Tsing Hua University. Article on Poncelet's Porism at Mathworld. Conic sections Elliptic curves