The conductance of quantum dots with ballistic point contacts supporti
ng a single channel and placed at arbitrary sites on their perimeters
has been numerically computed. The results indicate that whenever the
contacts break the dot symmetry, the statistics of the conductance clo
sely follow the circular orthogonal ensemble result, no matter whether
the shape of the quantum dot is itself regular or not and independent
ly of its size. On the other hand, the statistics of adjacent resonanc
es (a property closely related to the level statistics of the isolated
quantum dot) follows the Wigner-Dyson distribution only for small sys
tems. [S0163-1829(98)50540-1].