  We analyze the extraction of weak phases from CP violation in
$B\to\pi^+\pi^-$ decays. We propose to determine the unitarity triangle
$(\bar\rho,\bar\eta)$ by combining the information on mixing induced CP
violation in $B\to\pi^+\pi^-$, $S$, with the precision observable $\sin 2\beta$
obtained from the CP asymmetry in $B\to\psi K_S$. It is then possible to write
down exact analytical expressions for $\bar\rho$ and $\bar\eta$ as simple
functions of the observables $S$ and $\sin 2\beta$, and of the penguin
parameters $r$ and $\phi$. As an application clean lower bounds on $\bar\eta$
and $1-\bar\rho$ can be derived as functions of $S$ and $\sin 2\beta$,
essentially without hadronic uncertainty. Computing $r$ and $\phi$ within QCD
factorization yields precise determinations of $\bar\rho$ and $\bar\eta$ since
the dependence on $r$ and $\phi$ is rather weak. It is emphasized that the
sensitivity to the phase $\phi$ enters only at second order and is extremely
small for moderate values of this phase, predicted in the heavy-quark limit.
Transparent analytical formulas are further given and discussed for the
parameter $C$ of direct CP violation in $B\to\pi^+\pi^-$. We also discuss
alternative ways to analyze $S$ and $C$ that can be useful if new physics
affects $B_d$--$\bar B_d$ mixing. Predictions and uncertainties for $r$ and
$\phi$ in QCD factorization are examined in detail. It is pointed out that a
simultaneous expansion in $1/m_b$ and 1/N leads to interesting simplifications.
At first order infrared divergences are absent, while the most important
effects are retained. Independent experimental tests of the factorization
framework are briefly discussed.
