  $Z_1(4050)$ and $Z_2(4250)$ observed in $\bar{B}^0\to\chi_{c1}K^-\pi^+$ by
the Belle Collaboration are candidates of charged charmonium-like states that
minimally includes two quarks and two antiquarks. While $Z_1(4050)$ and
$Z_2(4250)$ have been interpreted as tetraquark states previously, we propose a
completely different scenario based on a kinematical effect called the triangle
singularity. We demonstrate that the triangle singularities cause in the
$\chi_{c1}\pi^+$ invariant mass distribution resonance-like bumps that fit very
well the Belle data. If these bumps are simulated by the $Z_1(4050)$ and
$Z_2(4250)$ resonance excitations, the spin-parity of them are predicted to be
$1^-$ for $Z_1(4050)$ and $1^+$ or $1^-$ for $Z_2(4250)$. The bump
corresponding to $Z_1(4050)$ has a highly asymmetric shape, which the Belle
data exactly indicate. We show that the asymmetric shape originates from an
interplay between the triangle singularity and the opening of the
$X(3872)\pi^+$ channel near the triangle-singularity energy. This
characteristic lineshape could be used to discriminate different
interpretations of $Z_1(4050)$. An interesting prediction from interpretting
$Z_1(4050)$ and $Z_2(4250)$ as the triangle singularities is that similar bumps
caused by the same mechanisms possibly appear also in $\bar{B}^0\to J/\psi
K^-\pi^+$ data; the already observed $Z_c(4200)$ corresponds to $Z_2(4250)$ of
$J^P=1^+$.
