The Dirac-like cone dispersion induced by accidental degeneracy is demonstrated in two-dimensional dielectric photonic crystals (PCs) of core-shell rods arranged in square and triangular lattices. The Dirac-like point (DLP) is achieved at the center of the Brillouin zone with a threefold degenerate state having two bands forming a Dirac cone and a third flat band intersecting the cone at the same frequency. This degenerate state is accidental and exists only for particular set of parameters of the PC. These parameters have been adjusted to obtain accidentally induced DLP. When the core region has a higher refractive index than the cladding, the DLP has monopole and dipole field configurations. When the core region has a lower refractive index than the cladding, the DLP is formed by quadrupole and dipole field configurations. The PCs exhibiting Dirac-like dispersion formed by monopole and dipole interactions can be mapped to an effective medium with and equal to zero. The isotropic behavior of DLP formed by monopole and dipole interactions is observed by computing isofrequency contours.