We study the problem of the exact computation of the medial axis of planar shapes the boundary of which is defined by piecewise conic arcs. The algorithm used is a tracing algorithm, similar to existing numeric algorithms. We trace the medial axis edge by edge. Instead of keeping track of points on the medial axis, we are keeping track of the corresponding footpoints on the boundary curves, thus dealing with bisector curves in parametric space. We exploit some algebraic and geometric properties of the bisector curves that allow for efficient trimming and we represent bifurcation points via their associated footpoints on the boundary, as algebraic numbers. The algorithm computes the correct topology of the medial axis identifying bifurcation points of arbitrary degree.