  We investigate the Dirichlet solution for a semianalytic continuous function
on the boundary of a semianalytic bounded domain in the plane. We show that the
germ of the Dirichlet solution at a boundary point with angle greater than 0
lies in a certain quasianalytic class used by Ilyashenko in his work on
Hilbert's 16th problem. With this result we can prove that the Dirichlet
solution is definable in an o-minimal structure if the angle at a singular
boundary point of the domain is an irrational multiple of $\pi$.
