diff --git a/Triangulation_3/include/CGAL/Triangulation_segment_traverser_3.h b/Triangulation_3/include/CGAL/Triangulation_segment_traverser_3.h
index 013511ae32a..8e08e2f0447 100644
--- a/Triangulation_3/include/CGAL/Triangulation_segment_traverser_3.h
+++ b/Triangulation_3/include/CGAL/Triangulation_segment_traverser_3.h
@@ -26,6 +26,7 @@
 #include <CGAL/Triangulation_vertex_base_3.h>
 #include <CGAL/Triangulation_simplex_3.h>
 
+#include <boost/optional.hpp>
 
 // If defined, type casting is done statically,
 // reducing type-safety overhead.
@@ -913,8 +914,11 @@ public:
           _curr_simplex = Simplex_3();//should not be reached
         else if (triangulation()->is_infinite(chnext) && is_same_edge(get_edge(), e_exit))
           _curr_simplex = chnext;
-        else
-          _curr_simplex = shared_facet(get_edge(), e_exit);
+        else {
+          auto facet_opt = shared_facet(get_edge(), e_exit);
+          if(static_cast<bool>(facet_opt)) _curr_simplex = *facet_opt;
+          else _curr_simplex = shared_cell(get_edge(), e_exit);
+        }
         break;
       }
 
@@ -1204,7 +1208,7 @@ private:
         && edge_has_vertex(e1, e2.first->vertex(e2.third));
   }
 
-  Vertex_handle shared_vertex(const Edge& e1, const Edge& e2) const
+  std::optional<Vertex_handle> shared_vertex(const Edge& e1, const Edge& e2) const
   {
     Vertex_handle v1a = e1.first->vertex(e1.second);
     Vertex_handle v1b = e1.first->vertex(e1.third);
@@ -1215,18 +1219,18 @@ private:
       return v1a;
     else if (v1b == v2a || v1b == v2b)
       return v1b;
-
-    std::cerr << "There is no vertex shared by e1 and e2" << std::endl;
-    CGAL_unreachable();
-    return Vertex_handle();
+    else
+      return {};
   }
 
-  Facet shared_facet(const Edge& e1, const Edge& e2) const
+  std::optional<Facet> shared_facet(const Edge& e1, const Edge& e2) const
   {
     Vertex_handle v2a = e2.first->vertex(e2.second);
     Vertex_handle v2b = e2.first->vertex(e2.third);
 
-    Vertex_handle sv = shared_vertex(e1, e2);
+    auto sv_opt = shared_vertex(e1, e2);
+    if(!sv_opt) return {};
+    Vertex_handle sv = *sv_opt;
     Vertex_handle nsv2 = (sv == v2a) ? v2b : v2a;
 
     typename Tr::Facet_circulator circ
@@ -1242,9 +1246,7 @@ private:
       }
     } while (++circ != end);
 
-    std::cerr << "There is no facet shared by e1 and e2" << std::endl;
-    CGAL_unreachable();
-    return Facet(Cell_handle(), 0);
+    return {};
   }
 
   Facet shared_facet(const Edge& e, const Vertex_handle v) const
@@ -1295,6 +1297,11 @@ private:
     }
   }
 
+  Cell_handle shared_cell(const Edge e1, const Edge e2) const {
+    auto facet = shared_facet(e1, e2.first->vertex(e2.second));
+    return shared_cell(facet, e2.first->vertex(e2.third));
+  }
+
 };//class Triangulation_segment_simplex_iterator_3
 
 } // namespace CGAL
diff --git a/Triangulation_3/test/Triangulation_3/test_segment_simplex_traverser_3.cpp b/Triangulation_3/test/Triangulation_3/test_segment_simplex_traverser_3.cpp
index 111eb35907c..d491066a4e8 100644
--- a/Triangulation_3/test/Triangulation_3/test_segment_simplex_traverser_3.cpp
+++ b/Triangulation_3/test/Triangulation_3/test_segment_simplex_traverser_3.cpp
@@ -373,18 +373,17 @@ bool test_a_simple_tetrahedron() {
   test({-.1, .5,  0}, {.5 , .5,  0}, "I121");
   test({-.1, .5,  0}, {.25, .5,  0}, "I12");
 
-
   // x [130] (entering by an edge and exiting by a non-incident vertex) is not possible
   //   because that would have passed by the common face -> see [120] ([021] in reverse)
 
   // [131] entering by an edge and exiting by the opposite edge in the cell,
   // on the line x==y==0.5-z, also known as (.5-z, .5-z, z)
-  test({  0,    0,    .5 }, { .25,  .25,  .25},   "13");
-  test({  0,    0,    .5 }, { .5,   .5,   0  },   "131");
-  test({  0,    0,    .5 }, { .55,  .55, -.05},   "131I");
-  test({ -.05, -.05,  .55}, { .55,  .55, -.05},  "I131I");
-  test({ -.05, -.05,  .55}, { .5 ,  .5 ,  0  },  "I131");
-  test({ -.05, -.05,  .55}, { .2 ,  .2 ,  .3 },  "I13");
+  test({  0,    0,    .5 }, { .25,  .25,   .25 },   "13");
+  test({  0,    0,    .5 }, { .5,   .5,    0   },   "131");
+  test({  0,    0,    .5 }, { .625, .625, -.125},   "131I");
+  test({ -.05, -.05,  .55}, { .625, .625, -.125},  "I131I");
+  test({ -.05, -.05,  .55}, { .5 ,  .5 ,   0   },  "I131");
+  test({ -.05, -.05,  .55}, { .25,  .25,   .25 },  "I13");
 
   // [132] queries entering by an edge and exiting by a facet, on the line (x, .25-x, x)
   test({  0, .25,  0}, { .20, .05,  .20},  "13");