diff --git a/src/kernel/topology/faces.rs b/src/kernel/topology/faces.rs
index 0e1eeb099..43fd42275 100644
--- a/src/kernel/topology/faces.rs
+++ b/src/kernel/topology/faces.rs
@@ -108,18 +108,41 @@ impl Face {
                 let outside = aabb.maxs * 2.;
 
                 triangles.retain(|triangle| {
-                    for segment in triangle.edges() {
+                    'outer: for segment in triangle.edges() {
                         let mut inverted_segment = segment;
                         inverted_segment.swap();
 
-                        let triangle_segment_is_face_edge = face_as_polygon
-                            .contains(&segment)
-                            || face_as_polygon.contains(&inverted_segment);
-
                         // If the segment is an edge of the face, we don't need
                         // to take a closer look.
-                        if triangle_segment_is_face_edge {
-                            continue;
+                        //
+                        // We can't use `contains` here, as that compares the
+                        // segments directly, without taking floating point
+                        // accuracy into account.
+                        //
+                        // This is not great. See this issue:
+                        // https://github.com/hannobraun/Fornjot/issues/78
+                        for s in &face_as_polygon {
+                            // This epsilon value is small enough to not mistake
+                            // different vertices as equal, for common use
+                            // cases, while still being a few orders of
+                            // magnitude larger than the differences between
+                            // equal vertices that I've seen so far.
+                            let eps = 1e-12;
+
+                            let aa = (s.a - segment.a).magnitude();
+                            let bb = (s.b - segment.b).magnitude();
+                            let ab = (s.a - segment.b).magnitude();
+                            let ba = (s.b - segment.a).magnitude();
+
+                            let segment_is_face_edge = aa < eps && bb < eps;
+                            let segment_is_inverted_face_edge =
+                                ab < eps && ba < eps;
+
+                            if segment_is_face_edge
+                                || segment_is_inverted_face_edge
+                            {
+                                continue 'outer;
+                            }
                         }
 
                         // To determine if the edge is within the polygon, we