On a circumference, points $A$ and $B$ are on opposite arcs of diameter $CD$. Line segments $CE$ and $DF$ are perpendicular to $AB$ such that $A-E-F-B$ (i.e., $A$, $E$, $F$ and $B$ are collinear on this order). Knowing $AE=1$, find the length of $BF$.
1. **Understanding the Problem:**
   - We have a circle with diameter \(CD\).
   - Points \(A\) and \(B\) are on opposite arcs of the diameter \(CD\).
   - Line segments \(CE\) and \(DF\) are perpendicular to \(AB\) such that \(A, E, F, B\) are collinear in this order.
   - Given \(AE = 1\), we need to find the length of \(BF\).

2. **Constructing the Geometry:**
   - Extend \(DE\) to meet the circumference again at \(D'\).
   - Extend \(CF\) to meet the circumference again at \(C'\).

3. **Analyzing the Arcs:**
   - Since \(DE\) and \(CF\) are perpendicular to \(AB\), they are radii of the circle.
   - The arcs \(AD'\) and \(BC\) are equal because they subtend equal angles at the circumference.
   - Similarly, the arcs \(BC'\) and \(AD\) are equal.

4. **Congruence of Triangles:**
   - Consider triangles \(\triangle DAD'\) and \(\triangle C'BC\).
   - These triangles are congruent because they have equal corresponding arcs and angles.
   - Therefore, the altitudes from \(A\) and \(B\) to \(AB\) are congruent.

5. **Conclusion:**
   - Since the altitudes from \(A\) and \(B\) are congruent, \(BF = AE\).
   - Given \(AE = 1\), it follows that \(BF = 1\).

The final answer is \(\boxed{1}\).