diff --git a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h
index 7b7dd93be68..ca771fe714a 100644
--- a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h
+++ b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h
@@ -2165,10 +2165,10 @@ Periodic_3_triangulation_3<GT,TDS>::periodic_insert(
     Point_hider &hider, Vertex_handle vh)
 {
   Vertex_handle v;
-  CGAL_triangulation_assertion(number_of_vertices() != 0);
+  CGAL_triangulation_precondition(number_of_vertices() != 0);
   CGAL_triangulation_precondition_code(
       Locate_type lt_assert; int i_assert; int j_assert;);
-  CGAL_triangulation_assertion(side_of_cell(tester.point(),o, c,
+  CGAL_triangulation_precondition(side_of_cell(tester.point(),o, c,
       lt_assert, i_assert, j_assert) != ON_UNBOUNDED_SIDE);
 
   tester.set_offset(o);
diff --git a/TDS_2/include/CGAL/Triangulation_data_structure_2.h b/TDS_2/include/CGAL/Triangulation_data_structure_2.h
index 9171f0e7fdf..d5dc2463b09 100644
--- a/TDS_2/include/CGAL/Triangulation_data_structure_2.h
+++ b/TDS_2/include/CGAL/Triangulation_data_structure_2.h
@@ -1735,7 +1735,7 @@ join_vertices(Face_handle f, int i, Vertex_handle v)
   if ( v1->face() == f || v1->face() == g ) v1->set_face(tl);
 
 
-#ifndef CGAL_NO_ASSERTIONS
+#if ! defined(CGAL_TRIANGULATION_NO_ASSERTIONS) && ! defined(CGAL_NO_ASSERTIONS)
   for (Face_iterator fit = faces_begin(); fit != faces_end(); ++fit) {
     int id;
     CGAL_triangulation_assertion( !fit->has_vertex(v2, id) );