The problem of intersecting two ellipses arises as a frequent subproblem in computational kinematics and geometry. In this paper, an efficient solution method to this problem is presented using the concept of the power of a point with respect to an ellipse. The point-ellipse power appears in Distance Geometry as a generalization to the squared distance between two points. For establishing the intersection method, several algebraic forms of ellipses are reviewed and the interoperability of distinct definitions for the power of points and ellipses is outlined.