Two critical points have been revealed in the normal-state phase diagram of the electron-doped cuprate superconductor Nd$_{2-x}$Ce$_x$CuO$_4$ by exploring the Fermi surface properties of high quality single crystals by high-field magnetotransport. First, the quantitative analysis of the Shubnikov-de Haas effect shows that the weak superlattice potential responsible for the Fermi surface reconstruction in the overdoped regime extrapolates to zero at the doping level $x_c = 0.175$ corresponding to the onset of superconductivity. Second, the high-field Hall coefficient exhibits a sharp drop right below optimal doping $x_{\mathrm{opt}} = 0.145$ where the superconducting transition temperature is maximum. This drop is most likely associated with the onset of long-range antiferromagnetic ordering. Thus, for Nd$_{2-x}$Ce$_x$CuO$_4$ the superconducting dome appears to be pinned by two critical points to the normal state phase diagram. ; Comment: 5 pages; 4 eps figures; Suppl. Material is available upon request