Determination of CKM phases through rigid polygons of flavor SU(3) amplitudes

Some new methods for the extraction of CKM phases $\alpha$ and $\gamma$ using flavor SU(3) symmetry are suggested. Rigid polygons are constructed in the complex plane with sides equal to the decay amplitudes of $B$ mesons into two light (charmless) pseudoscalar mesons. These rigid polygons incorporate all the possible amplitude triangles and, being overdetermined, also serve as consistency checks and in estimating the rates of some decay modes. The same techniques also lead to numerous useful amplitude triangles when octet-singlet mixing has been taken into account and nearly physical $\eta$, ${\eta }^{\prime }$ are used.