Based on the Fraunhofer diffraction integral formula, the propagation expressions for radially polarized beam diffracted by a triangular multipoint plate (TMP) are derived. Through the theoretical calculation and experiments, the intensity distribution in the diffraction field arranged periodically is got. The polarization singularities of radially polarized beam in the diffraction field by using the complex Stokes fields is analyzed. It is shown that the light intensity patterns varies with the N-value of the TMP. The number of the polarization singularities increases with the N-value of the TMP. The beam parameters (the wavelength and the waist width of the beam), transmission distance and the parameters of the TMP (the size of each side of the triangle) have influences on the positions of the polarization singularities. The creation and annihilation of the polarization singularities may appear as the N-value of the TMP varies.