diff --git a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2.h b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2.h
index 4918683b6b7..79103211ffd 100644
--- a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2.h
+++ b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2.h
@@ -1657,12 +1657,14 @@ protected:
   }
 
   inline Oriented_side
-  oriented_side(const Storage_site_2& q, const Storage_site_2& supp,
+  oriented_side(const Storage_site_2& s1, const Storage_site_2& s2,
+                const Storage_site_2& supp,
 		const Storage_site_2& p) const
   {
-    CGAL_precondition( q.is_point() && supp.is_segment() && p.is_point() );
+    CGAL_precondition( supp.is_segment() && p.is_point() );
     return
-      geom_traits().oriented_side_2_object()(q.site(), supp.site(), p.site());
+      geom_traits().oriented_side_2_object()(
+          s1.site(), s2.site(), supp.site(), p.site());
   }
 
   inline Oriented_side
@@ -1869,10 +1871,11 @@ protected:
   }
 
   inline Oriented_side
-  oriented_side(const Site_2& q, const Site_2& supp, const Site_2& p) const
+  oriented_side(const Site_2& s1, const Site_2&s2,
+                const Site_2& supp, const Site_2& p) const
   {
-    CGAL_precondition( q.is_point() && supp.is_segment() && p.is_point() );
-    return geom_traits().oriented_side_2_object()(q, supp, p);
+    CGAL_precondition( supp.is_segment() && p.is_point() );
+    return geom_traits().oriented_side_2_object()(s1, s2, supp, p);
   }
 
   inline Oriented_side
diff --git a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Oriented_side_C2.h b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Oriented_side_C2.h
index 1dd0fde727c..80f55ac41b5 100644
--- a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Oriented_side_C2.h
+++ b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Oriented_side_C2.h
@@ -60,6 +60,19 @@ public:
   typedef Oriented_side                       result_type;
   typedef Site_2                              argument_type;
 
+  // computes the oriented side of the Voronoi vertex of s1, s2, inf
+  // wrt the line that passes through the point p and its direction
+  // is the direction of the supporting line of s, rotated by 90
+  // degrees counterclockwise.
+  Oriented_side operator()(const Site_2& s1, const Site_2& s2,
+			   const Site_2& s, const Site_2& p) const
+  {
+    CGAL_precondition(s1.is_point() or s2.is_point());
+    CGAL_precondition(s1.is_segment() or s2.is_segment());
+
+    return this->operator()((s1.is_point()? s1: s2), s, p);
+  }
+
   // computes the oriented side of the point q
   // wrt the line that is passes through the point p and its direction
   // is the direction of the supporting line of s, rotated by 90
diff --git a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Segment_Delaunay_graph_2_impl.h b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Segment_Delaunay_graph_2_impl.h
index 12eef09d243..247470037f0 100644
--- a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Segment_Delaunay_graph_2_impl.h
+++ b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Segment_Delaunay_graph_2_impl.h
@@ -527,13 +527,13 @@ find_faces_to_split(const Vertex_handle& v, const Site_2& t) const
 	CGAL_assertion(  !is_infinite( ff1->vertex(ccw_v) )  );
 	CGAL_assertion( ff1->vertex(ccw_v)->site().is_point() );
 	sv_ep = ff1->vertex(ccw_v)->site();
+        os1 = oriented_side(v->site(), sv_ep, sitev_supp, t);
       } else {
 	CGAL_assertion(  !is_infinite( ff1->vertex( cw_v) )  );
 	CGAL_assertion( ff1->vertex( cw_v)->site().is_point() );
 	sv_ep = ff1->vertex( cw_v)->site();
+        os1 = oriented_side(sv_ep, v->site(), sitev_supp, t);
       }
-
-      os1 = oriented_side(sv_ep, sitev_supp, t);
     } else {
       os1 = oriented_side(fc1->vertex(0)->site(),
 			  fc1->vertex(1)->site(),
@@ -551,13 +551,13 @@ find_faces_to_split(const Vertex_handle& v, const Site_2& t) const
 	CGAL_assertion(  !is_infinite( ff2->vertex(ccw_v) )  );
 	CGAL_assertion( ff2->vertex(ccw_v)->site().is_point() );
 	sv_ep = ff2->vertex(ccw_v)->site();
+        os2 = oriented_side(v->site(), sv_ep, sitev_supp, t);
       } else {
 	CGAL_assertion(  !is_infinite( ff2->vertex( cw_v) )  );
 	CGAL_assertion( ff2->vertex( cw_v)->site().is_point() );
 	sv_ep = ff2->vertex( cw_v)->site();
+        os2 = oriented_side(sv_ep, v->site(), sitev_supp, t);
       }
-
-      os2 = oriented_side(sv_ep, sitev_supp, t);
     } else {
       os2 = oriented_side(fc2->vertex(0)->site(),
 			  fc2->vertex(1)->site(),