diff --git a/Filtered_kernel/include/CGAL/Filtered_kernel.h b/Filtered_kernel/include/CGAL/Filtered_kernel.h
index 81b8bd4d07d..06a9fa53829 100644
--- a/Filtered_kernel/include/CGAL/Filtered_kernel.h
+++ b/Filtered_kernel/include/CGAL/Filtered_kernel.h
@@ -61,6 +61,8 @@ struct Filtered_kernel_base
     enum { Has_filtered_predicates = true };
     typedef Boolean_tag<Has_filtered_predicates> Has_filtered_predicates_tag;
 
+    typedef Tag_true Can_construct_exact_intersection_point_2;
+
     template < typename Kernel2 >
     struct Base {
         typedef typename CK::template Base<Kernel2> CK2;
diff --git a/Installation/include/CGAL/Exact_kernel_selector_fwd.h b/Installation/include/CGAL/Exact_kernel_selector_fwd.h
new file mode 100644
index 00000000000..8f0213606a7
--- /dev/null
+++ b/Installation/include/CGAL/Exact_kernel_selector_fwd.h
@@ -0,0 +1,25 @@
+// Copyright (c) 2017
+// Utrecht University (The Netherlands),
+// ETH Zurich (Switzerland),
+// INRIA Sophia-Antipolis (France),
+// Max-Planck-Institute Saarbruecken (Germany),
+// and Tel-Aviv University (Israel).  All rights reserved.
+//
+// This file is part of CGAL (www.cgal.org)
+//
+// $URL$
+// $Id$
+// SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial
+//
+// Author(s)     : Sylvain Pion,
+//                 Mael Rouxel-Labbé
+
+#ifndef CGAL_EXACT_KERNEL_SELECTOR_FWD_H
+#define CGAL_EXACT_KERNEL_SELECTOR_FWD_H
+
+namespace CGAL {
+template <class CK, class Rep = typename CK::Rep_tag /* Cartesian_tag */>
+struct Exact_kernel_selector;
+}
+
+#endif // CGAL_EXACT_KERNEL_SELECTOR_FWD_H
diff --git a/Intersections_2/include/CGAL/Intersections_2/Segment_2_Segment_2.h b/Intersections_2/include/CGAL/Intersections_2/Segment_2_Segment_2.h
index 19d1b300232..e7725ea561c 100644
--- a/Intersections_2/include/CGAL/Intersections_2/Segment_2_Segment_2.h
+++ b/Intersections_2/include/CGAL/Intersections_2/Segment_2_Segment_2.h
@@ -27,6 +27,8 @@
 #include <CGAL/Intersections_2/Line_2_Line_2.h>
 #include <CGAL/Uncertain.h>
 #include <CGAL/Intersection_traits_2.h>
+#include <CGAL/Cartesian_converter.h>
+#include <CGAL/Exact_kernel_selector_fwd.h>
 
 namespace CGAL {
 
@@ -456,6 +458,33 @@ intersection(const typename K::Segment_2 &seg1,
     }
 }
 
+template <class K>
+boost::optional<typename K::Point_2>
+exact_intersection_point(const typename K::Point_2& pa,
+                         const typename K::Point_2& pb,
+                         const typename K::Point_2& pc,
+                         const typename K::Point_2& pd,
+                         const K&)
+{
+  using EK = typename Exact_kernel_selector<K>::Exact_kernel;
+  using EK_Line_2 = typename EK::Line_2;
+  using EK_Point_2 = typename EK::Point_2;
+  Cartesian_converter<K, EK> to_exact;
+  Cartesian_converter<EK, K> from_exact;
+  typename EK::Intersect_2 exact_intersect;
+  const auto exact_intersection =
+      exact_intersect(EK_Line_2{to_exact(pa), to_exact(pb)},
+                      EK_Line_2{to_exact(pc), to_exact(pd)});
+  if(!exact_intersection.has_value()) return {};
+  using boost::get;
+  if (const auto *p = get<EK_Point_2>(&*exact_intersection)) {
+    return { from_exact(*p) };
+  } else {
+    return {};
+  }
+
+}
+
 } // namespace internal
 } // namespace Intersections
 
@@ -464,4 +493,6 @@ CGAL_DO_INTERSECT_FUNCTION_SELF(Segment_2, 2)
 
 } //namespace CGAL
 
+#include <CGAL/Exact_kernel_selector.h>
+
 #endif
diff --git a/Kernel_23/include/CGAL/Exact_kernel_selector.h b/Kernel_23/include/CGAL/Exact_kernel_selector.h
index 9929cb5a02b..23ed23d3afb 100644
--- a/Kernel_23/include/CGAL/Exact_kernel_selector.h
+++ b/Kernel_23/include/CGAL/Exact_kernel_selector.h
@@ -29,9 +29,11 @@
 #include <CGAL/Cartesian_converter.h>
 #include <CGAL/Homogeneous_converter.h>
 
+#include <CGAL/Exact_kernel_selector_fwd.h>
+
 namespace CGAL {
 
-template <class CK, class Rep = typename CK::Rep_tag /* Cartesian_tag */>
+template <class CK, class Rep /* = typename CK::Rep_tag (Cartesian_tag) */>
 struct Exact_kernel_selector
 {
   typedef typename internal::Exact_field_selector<typename CK::RT>::Type  Exact_nt;
diff --git a/Kernel_23/include/CGAL/Kernel/function_objects.h b/Kernel_23/include/CGAL/Kernel/function_objects.h
index 2bc9704ed04..6c00b7591aa 100644
--- a/Kernel_23/include/CGAL/Kernel/function_objects.h
+++ b/Kernel_23/include/CGAL/Kernel/function_objects.h
@@ -3553,6 +3553,17 @@ namespace CommonKernelFunctors {
     { return Intersections::internal::intersection(t1, t2, K()); }
   };
 
+  template <typename K>
+  class Construct_exact_intersection_point_2
+  {
+  public:
+
+    template <class... T>
+    decltype(auto)
+    operator()(const T&... t) const
+    { return Intersections::internal::exact_intersection_point(t..., K()); }
+  };
+
   template <typename K>
   class Intersect_3
   {
diff --git a/Kernel_23/include/CGAL/Kernel/interface_macros.h b/Kernel_23/include/CGAL/Kernel/interface_macros.h
index d0ba827fa37..e0d6c57bbf6 100644
--- a/Kernel_23/include/CGAL/Kernel/interface_macros.h
+++ b/Kernel_23/include/CGAL/Kernel/interface_macros.h
@@ -302,6 +302,8 @@ CGAL_Kernel_cons(Construct_circumcenter_3,
                  construct_circumcenter_3_object)
 CGAL_Kernel_cons(Construct_weighted_circumcenter_3,
                  construct_weighted_circumcenter_3_object)
+CGAL_Kernel_cons(Construct_exact_intersection_point_2,
+                 construct_exact_intersection_point_2_object)
 CGAL_Kernel_cons(Compute_power_product_2,
                  compute_power_product_2_object)
 CGAL_Kernel_cons(Compute_power_product_3,
diff --git a/Triangulation_2/include/CGAL/Constrained_triangulation_2.h b/Triangulation_2/include/CGAL/Constrained_triangulation_2.h
index 57c36a5ee36..9f8f515eccd 100644
--- a/Triangulation_2/include/CGAL/Constrained_triangulation_2.h
+++ b/Triangulation_2/include/CGAL/Constrained_triangulation_2.h
@@ -28,10 +28,11 @@
 #include <CGAL/Default.h>
 #include <CGAL/intersections.h>
 #include <CGAL/squared_distance_2.h>
-#include <CGAL/Exact_rational.h>
-#include <CGAL/Simple_cartesian.h>
+#include <CGAL/tags.h>
+
 
 #include <boost/mpl/if.hpp>
+#include <boost/mpl/has_xxx.hpp>
 #include <boost/iterator/filter_iterator.hpp>
 
 #include <boost/utility/result_of.hpp>
@@ -1132,25 +1133,13 @@ intersect(Face_handle f, int i,
     }
   }
   else{ //intersection computed
-    if(intersection_not_in_the_two_triangle(pi)) {
+    if(can_construct_almost_exact_intersection(geom_traits(), itag) &&
+       intersection_not_in_the_two_triangle(pi))
+    {
       // now compute the exact intersection point
-      using EK = Simple_cartesian<Exact_rational>;
-      using EK_Line_2  = EK::Line_2;
-      using EK_Point_2 = EK::Point_2;
-      Cartesian_converter<Gt, EK> to_exact;
-      Cartesian_converter<EK, Gt> from_exact;
-      EK::Intersect_2 exact_intersect;
-      auto exact_intersection =
-          exact_intersect(EK_Line_2{to_exact(pa), to_exact(pb)},
-                          EK_Line_2{to_exact(pc), to_exact(pd)});
-      CGAL_assertion(exact_intersection.has_value());
-      using boost::get;
-      if (const auto *p = get<EK_Point_2>(&*exact_intersection)) {
-        pi = from_exact(*p);
-      } else {
-        CGAL_error();
-      }
-      if(intersection_not_in_the_two_triangle(pi)) {
+      const bool result =
+          almost_exact_intersection(geom_traits(), pa, pb, pc, pd, pi, itag);
+      if (!result || intersection_not_in_the_two_triangle(pi)) {
         // If the most-exact intersection point is not in the union of the two
         // triangles, then snap to `pc` or `pd`...
         if(compare_distance(pi, pc, pd) == SMALLER) {
@@ -1831,11 +1820,11 @@ limit_intersection(const Gt& ,
 template<class Gt>
 int
 limit_intersection(const Gt& gt,
-             const typename Gt::Point_2& pa,
-             const typename Gt::Point_2& pb,
-             const typename Gt::Point_2& pc,
-             const typename Gt::Point_2& pd,
-             Exact_predicates_tag)
+                   const typename Gt::Point_2& pa,
+                   const typename Gt::Point_2& pb,
+                   const typename Gt::Point_2& pc,
+                   const typename Gt::Point_2& pd,
+                   Exact_predicates_tag)
 {
   typename Gt::Construct_line_2 line = gt.construct_line_2_object();
   typename Gt::Compute_squared_distance_2
@@ -1853,6 +1842,82 @@ limit_intersection(const Gt& gt,
   return i;
 }
 
+BOOST_MPL_HAS_XXX_TRAIT_DEF(Can_construct_exact_intersection_point_2)
+
+template <typename Gt,
+          bool = has_Can_construct_exact_intersection_point_2<Gt>::value>
+struct Can_construct_almost_exact_intersection
+    : public CGAL::Boolean_tag<
+          Gt::Can_construct_exact_intersection_point_2::value> {};
+
+template <typename Gt>
+struct Can_construct_almost_exact_intersection<Gt, false>
+    : public CGAL::Tag_false {};
+
+template <typename Gt>
+constexpr bool can_construct_almost_exact_intersection(const Gt&, Exact_predicates_tag) {
+  return Can_construct_almost_exact_intersection<Gt>::value;
+}
+
+template <typename Gt, typename Tag>
+constexpr bool can_construct_almost_exact_intersection(const Gt&, Tag) {
+  return false;
+}
+
+template <typename Gt, typename Tag>
+bool almost_exact_intersection(const Gt&,
+                               const typename Gt::Point_2&,
+                               const typename Gt::Point_2&,
+                               const typename Gt::Point_2&,
+                               const typename Gt::Point_2&,
+                               typename Gt::Point_2&,
+                               Tag)
+{
+  CGAL_error_msg("this function should be call only with Exact_predicates_tag");
+}
+
+template <typename Gt>
+bool almost_exact_intersection(const Gt& gt,
+                               const typename Gt::Point_2& pa,
+                               const typename Gt::Point_2& pb,
+                               const typename Gt::Point_2& pc,
+                               const typename Gt::Point_2& pd,
+                               typename Gt::Point_2& pi,
+                               Exact_predicates_tag)
+{
+  Boolean_tag<Can_construct_almost_exact_intersection<Gt>::value> tag;
+  return almost_exact_intersection(gt, pa, pb, pc, pd, pi, tag);
+}
+
+template <typename Gt>
+bool almost_exact_intersection(const Gt&,
+                               const typename Gt::Point_2&,
+                               const typename Gt::Point_2&,
+                               const typename Gt::Point_2&,
+                               const typename Gt::Point_2&,
+                               typename Gt::Point_2&,
+                               Tag_false)
+{
+  CGAL_error_msg("this function should be call only with Exact_predicates_tag"
+                 " and with an appropriate traits class");
+}
+
+template <typename Gt>
+bool almost_exact_intersection(const Gt& gt,
+                               const typename Gt::Point_2& pa,
+                               const typename Gt::Point_2& pb,
+                               const typename Gt::Point_2& pc,
+                               const typename Gt::Point_2& pd,
+                               typename Gt::Point_2& pi,
+                               Tag_true /* gt has Construct_exact_intersection_point_2 */)
+{
+  auto exact_intersect = gt.construct_exact_intersection_point_2_object();
+  const auto exact_intersection = exact_intersect(pa, pb, pc, pd);
+  if(!exact_intersection.has_value()) return false;
+  pi = *exact_intersection;
+  return true;
+}
+
 } //namespace CGAL
 
 #include <CGAL/enable_warnings.h>