diff --git a/Intersections_3/include/CGAL/Intersections_3/internal/intersection_3_1_impl.h b/Intersections_3/include/CGAL/Intersections_3/internal/intersection_3_1_impl.h
index 8206961896d..a9066d1373c 100644
--- a/Intersections_3/include/CGAL/Intersections_3/internal/intersection_3_1_impl.h
+++ b/Intersections_3/include/CGAL/Intersections_3/internal/intersection_3_1_impl.h
@@ -223,31 +223,55 @@ intersection_point(const typename K::Plane_3 &plane1,
     const FT &m00 = plane1.a();
     const FT &m01 = plane1.b();
     const FT &m02 = plane1.c();
-    const FT &b0 =  plane1.d();
+    const FT &b0  = - plane1.d();
     const FT &m10 = plane2.a();
     const FT &m11 = plane2.b();
     const FT &m12 = plane2.c();
-    const FT &b1 =  plane2.d();
+    const FT &b1  = - plane2.d();
     const FT &m20 = plane3.a();
     const FT &m21 = plane3.b();
     const FT &m22 = plane3.c();
-    const FT &b2 = plane3.d();
-
-    const FT den = - determinant(m00, m01, m02,
-                                 m10, m11, m12,
-                                 m20, m21, m22);
+    const FT &b2  = - plane3.d();
+    /*
+    const FT den = determinant(m00, m01, m02,
+                               m10, m11, m12,
+                               m20, m21, m22);
     if(den == FT(0)){
       return boost::none;
     }
     const FT num1 = determinant(b0, m01, m02,
-                               b1, m11, m12,
-                               b2, m21, m22);
+                                b1, m11, m12,
+                                b2, m21, m22);
     const FT num2 = determinant(m00, b0, m02,
-                               m10, b1, m12,
-                               m20, b2, m22);
+                                m10, b1, m12,
+                                m20, b2, m22);
     const FT num3 = determinant(m00, m01, b0,
-                               m10, m11, b1,
-                               m20, m21, b2);
+                                m10, m11, b1,
+                                m20, m21, b2);
+    */
+
+    // Minors common to two determinants
+    const FT minor_0 = m00*m11 - m10*m01;
+    const FT minor_1 = m00*m21 - m20*m01;
+    const FT minor_2 = m10*m21 - m20*m11;
+
+    const FT den = minor_0*m22 - minor_1*m12 + minor_2*m02; // determinant of M
+
+    if(den == FT(0)){
+      return boost::none;
+    }
+
+    const FT num3 = minor_0*b2 - minor_1*b1 + minor_2*b0;  // determinant of M with M[x:2] swapped with [b0,b1,b2]
+
+    // Minors common to two determinants
+    const FT minor_3 = b0*m12 - b1*m02;
+    const FT minor_4 = b0*m22 - b2*m02;
+    const FT minor_5 = b1*m22 - b2*m12;
+
+    // num1 has opposite signs because b0 and M[:1] have been swapped
+    const FT num1 = - minor_3*m21 + minor_4*m11 - minor_5*m01;  // determinant of M with M[x:0] swapped with [b0,b1,b2]
+    const FT num2 = minor_3*m20 - minor_4*m10 + minor_5*m00;  // determinant of M with M[x:1] swapped with [b0,b1,b2]
+
     return boost::make_optional(typename K::Point_3(num1/den, num2/den, num3/den));
 }