diff --git a/Triangulation_3/include/CGAL/Conforming_Delaunay_triangulation_3.h b/Triangulation_3/include/CGAL/Conforming_Delaunay_triangulation_3.h
index 53fd471d450..e533992bb2d 100644
--- a/Triangulation_3/include/CGAL/Conforming_Delaunay_triangulation_3.h
+++ b/Triangulation_3/include/CGAL/Conforming_Delaunay_triangulation_3.h
@@ -161,7 +161,7 @@ protected:
     const T_3* tr;
     Compare_vertex_handle(const T_3* tr) : tr(tr) {}
     bool operator()(const Vertex_handle va, const Vertex_handle vb) const {
-      return tr->compare_xyz(tr->point(va), tr->point(vb)) == SMALLER;
+      return va < vb;
     }
   };
 
diff --git a/Triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_3.h b/Triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_3.h
index d2d47bdcb83..c347e3f9c47 100644
--- a/Triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_3.h
+++ b/Triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_3.h
@@ -1530,12 +1530,18 @@ private:
               auto tetrahedron =
                   typename Geom_traits::Tetrahedron_3{tr.point(n_ch->vertex(0)), tr.point(n_ch->vertex(1)),
                                                       tr.point(n_ch->vertex(2)), tr.point(n_ch->vertex(3))};
+              auto tet_bbox = tetrahedron.bbox();
               if(std::any_of(fh_region.begin(), fh_region.end(), [&](auto fh) {
                    const auto v0 = fh->vertex(0)->info().vertex_handle_3d;
                    const auto v1 = fh->vertex(1)->info().vertex_handle_3d;
                    const auto v2 = fh->vertex(2)->info().vertex_handle_3d;
                    const auto triangle = typename Geom_traits::Triangle_3{tr.point(v0), tr.point(v1), tr.point(v2)};
-                   return does_tetrahedron_intersect_triangle_interior(tetrahedron, triangle, tr.geom_traits());
+                   const auto tri_bbox = triangle.bbox();
+                   if(CGAL::do_overlap(tet_bbox, tri_bbox)) {
+                     return does_tetrahedron_intersect_triangle_interior(tetrahedron, triangle, tr.geom_traits());
+                   } else {
+                     return false;
+                   }
                  }))
               {
                 intersecting_cells.insert(n_ch);