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A theoretical study of a waveguide with a uniformly curved section and an embedded quantum dot is presented within the envelope function approximation. For the quantum dot being extremely localized in the direction of the electron propagation, exact form of the scattering matrix of the system is derived, and a conductance of the waveguide is calculated. It is shown that if, for the straight wire with dot, in the fundamental mode conductance is a monotonically increasing function of the Fermi energy, then, after bending the waveguide, one has Fano resonances on the conductance-Fermi energy dependence. The resonances appear as a result of mixing by the bend of the longitudinal and transverse electron motion in the straight parts of the waveguide and concomitant intersubband interaction. For the small bend angle (large bend radius) the width of the resonance grows narrower with decreasing the angle (increasing the radius), until in the limit of the zero angle (infinite radius) one recovers true bound state in the continuum and the corresponding monotonic conductance of the straight channel. A dependence of the resonance width on the parameters of the bend and the dot is investigated; in particular, it is shown that, as a result of coherent resonant phenomena in the superposition of the bend and the dot, true bound states in the continuum can be formed also for the nonzero bend angle and the finite bend radius. Miscellaneous cases of bending the waveguide which, if uncurved, already exhibits the Fano resonance, are also investigated; for example, it is shown that in this situation, as a result of the bend, resonances can collapse too, again producing true bound states in the continuum. The case of inversion of the Fano resonances, i.e., a change of their minimum and maximum mutual location, is also analyzed. Mathematical and physical interpretation of the obtained results is given, and characteristic features of the critical parameters at which the Fano resonances collapse, are discussed. It is demonstrated that currents flowing in the waveguide, near the Fano resonances drastically change their behavior from laminar to vortical structure, and an evolution of the vortices is described. Parallels are drawn to the other types of the guiding structures, such as electromagnetic waveguides and acoustic ducts.