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$Z_1(4050)$ and $Z_2(4250)$ observed in $\bar{B}^0\to\chi_{c1}K^-\pi^+$ bythe Belle Collaboration are candidates of charged charmonium-like states thatminimally includes two quarks and two antiquarks. While $Z_1(4050)$ and$Z_2(4250)$ have been interpreted as tetraquark states previously, we propose acompletely different scenario based on a kinematical effect called the trianglesingularity. We demonstrate that the triangle singularities cause in the$\chi_{c1}\pi^+$ invariant mass distribution resonance-like bumps that fit verywell 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 bumpcorresponding to $Z_1(4050)$ has a highly asymmetric shape, which the Belledata exactly indicate. We show that the asymmetric shape originates from aninterplay between the triangle singularity and the opening of the$X(3872)\pi^+$ channel near the triangle-singularity energy. Thischaracteristic lineshape could be used to discriminate differentinterpretations of $Z_1(4050)$. An interesting prediction from interpretting$Z_1(4050)$ and $Z_2(4250)$ as the triangle singularities is that similar bumpscaused by the same mechanisms possibly appear also in $\bar{B}^0\to J/\psiK^-\pi^+$ data; the already observed $Z_c(4200)$ corresponds to $Z_2(4250)$ of$J^P=1^+$.

Triangle singularities in $\bar{B}^0\to \chi_{c1}K^-\pi^+$ relevant to $Z_1(4050)$ and $Z_2(4250)$