  An isospin analysis of $B\to\pi\pi$ decays yields $\sin 2\alpha$, where
$\alpha$ is the usual CKM angle $\alpha\equiv arg [-V_{td}V_{tb}^\ast
/(V_{ud}V_{ub}^\ast)]$, without hadronic uncertainty if isospin is a perfect
symmetry. Isospin, however, is broken not only by electroweak effects but also
by the u and d quark mass difference. The latter generates $\pi^0-\eta, \eta'$
mixing and converts the isospin-perfect triangle relation between the
$B\to\pi\pi$ amplitudes to a quadrilateral. The combined isospin-violating
effects impact the extracted value of $\sin 2\alpha$ in a significant manner,
particularly if the latter is small.
