When computing Monotone Polygon Triangulation, how do I store the formed diagonals in the DCEL?

It’s my understanding that a DCEL have the following structs

public class Vertex{   public Point Coordinate;   public Edge IncidentEdge; }  public class Edge{   public Vertex Origin;   public Edge Twin;   public Face IncidentFace;   public Edge Next;   public Edge Prev; }  public class Face{   public Edge IncidentEdge;   public List<Edge> edges; } 

If I go about determining the diagonals based on the type of vertex I’m at, how would I store the diagonal that was formed. Vertices can only store one incident edge. If I create a diagonal, any given vertex will have an additional incident edge. Do I just ignore this and just fix the half-edge pointers?