The problem of locating stagnation points in the flow produced by a system of N vortices in two dimensions is considered. The general solution follows from a 1864 theorem by Siebeck, that the stagnation points are the foci of a certain plane curve of class N-1 that has all lines connecting vortices pairwise as tangents. The case N=3, for which Siebeck's curve is a conic, is consdiered in some detail. It is shown that the classification of the type of conic coincides with the known classification of regimes of motin for the three vortices. A similarity result for the triangular coordinates of the stagnation point in a flow produced by three vortices with sum of strengths zero is found. Using topological arguments the distinct streamline patterns for flow about three vortices are also determined. Partial results are given for two special sets of vortex strengths on the changes between these patterns as the motion evolves. The analysis requires a number of unfamiliar mathematical tools which are explained.