We carry out numerical simulations of solidification of a two-dimensional liquid droplet cooled at a point with different conditions at the tri-junctions where the solid, liquid, and gas phases meet. The velocity of the advancing solidification front is determined from the quasi-steady heat conduction equations in both solid and liquid phases. The singularity in heat flux at the tri-junction encountered in the previous studies is shown to disappear when kinetic undercooling and the Gibbs-Thomson effect at the solid-liquid interface are taken into account. Locally regular solutions can be obtained for arbitrary values of the contact angles at the tri-junction. The solidification rate and shapes of solid particles are studied as functions of the surface tension, the contact angles, and the thermal conductivities of the two phases. Higher solidification rates are found for lower surface tensions and larger solid-to-liquid conductivity ratios. In the final stages of solidification, a corner or cusp is formed in the shape of the solid in agreement with related experimental observations. The corner angle is found to decrease when the solid-to-liquid density ratio is decreased. More elongated particles are obtained for higher surface tensions and lower ratios of densities, solid to liquid. (C) 2003 Elsevier B.V. All rights reserved.