A constant-work-space algorithm has read-only access to an input array and may use only O(1) additional words of bits, where n is the input size. We show how to triangulate a plane straight-line graph with n vertices in O(n2) time and constant work-space. We also consider the problem of preprocessing a simple polygon P for shortest path queries, where P is given by the ordered sequence of its n vertices. For this, we relax the space constraint to allow s words of work-space. After quadratic preprocessing, the shortest path between any two points inside P can be found in O(n2/s) time.