Poncelet MacBeath N=3 Conservations and Spherical Curve of the Sidelengths

Video shows that MacBeath Poncelet triangles conserve, besides the sum of squared sidelengths and the sum of squared cosines (or double angle cosines), the product of its cosines. Also, after a suggestion by Prof. Rich Schwartz, I am also showing the spherical curve swept by the sidelengths.