diff --git a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_Delaunay_triangulation_3.h b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_Delaunay_triangulation_3.h
index 5bfe6f65892..11ac33d2628 100644
--- a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_Delaunay_triangulation_3.h
+++ b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_Delaunay_triangulation_3.h
@@ -490,21 +490,27 @@ public:
       is_large_point_set = false;
 
     std::vector<Point> points(first, last);
-    CGAL::cpp98::random_shuffle (points.begin(), points.end());
-    Cell_handle hint;
     std::vector<Vertex_handle> dummy_points, double_vertices;
     typename std::vector<Point>::iterator pbegin = points.begin();
     if(is_large_point_set)
+    {
       dummy_points = insert_dummy_points();
-    else while(!is_1_cover()) {
-      insert(*pbegin);
-      ++pbegin;
-      if(pbegin == points.end())
-        return number_of_vertices() - n;
+    } else {
+      CGAL::cpp98::random_shuffle(points.begin(), points.end());
+      pbegin = points.begin();
+      while(!is_1_cover()) {
+        insert(*pbegin);
+        ++pbegin;
+        if(pbegin == points.end())
+          return number_of_vertices() - n;
+      }
     }
 
+    CGAL_postcondition(is_1_cover());
+
     spatial_sort (pbegin, points.end(), this->geom_traits());
 
+    Cell_handle hint;
     Conflict_tester tester(*pbegin,this);
     Point_hider hider;
     Cover_manager cover_manager(*this);
diff --git a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_regular_triangulation_3.h b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_regular_triangulation_3.h
index 14a7419274c..180cb4e36ee 100644
--- a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_regular_triangulation_3.h
+++ b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_regular_triangulation_3.h
@@ -519,8 +519,6 @@ public:
       is_large_point_set = false;
 
     std::vector<Weighted_point> points(first, last);
-    CGAL::cpp98::random_shuffle(points.begin(), points.end());
-    Cell_handle hint;
     std::vector<Vertex_handle> dummy_points_vhs, double_vertices;
     std::vector<Weighted_point> dummy_points;
     typename std::vector<Weighted_point>::iterator pbegin = points.begin();
@@ -533,6 +531,8 @@ public:
     }
     else
     {
+      CGAL::cpp98::random_shuffle(points.begin(), points.end());
+      pbegin = points.begin();
       while(!is_1_cover())
       {
         insert(*pbegin);
@@ -542,6 +542,8 @@ public:
       }
     }
 
+    CGAL_postcondition(is_1_cover());
+
     // Spatial sorting can only be applied to bare points, so we need an adaptor
     typedef typename Geom_traits::Construct_point_3 Construct_point_3;
     typedef typename boost::result_of<const Construct_point_3(const Weighted_point&)>::type Ret;
@@ -553,6 +555,7 @@ public:
                    CGAL::internal::boost_::make_function_property_map<Weighted_point, Ret, Construct_point_3>(
                        geom_traits().construct_point_3_object()), geom_traits()));
 
+    Cell_handle hint;
     Conflict_tester tester(*pbegin, this);
     Point_hider hider(this);
     Cover_manager cover_manager(*this);