Let a non-equilateral triangle $ ABC$ and $ AD,BE,CF$ are its altitudes. On the rays $ AD,BE,CF,$ respectively, let $ A',B',C'$ such that $ \frac {AA'}{AD} \equal{} \frac {BB'}{BE} \equal{} \frac {CC'}{CF} \equal{} k$. Find all values of $ k$ such that $ \triangle A'B'C'\sim\triangle ABC$ for any non-triangle $ ABC.$
1. **Given Setup:**
   - Let \( \triangle ABC \) be a non-equilateral triangle.
   - \( AD, BE, CF \) are the altitudes of \( \triangle ABC \).
   - Points \( A', B', C' \) are on the rays \( AD, BE, CF \) respectively such that \( \frac{AA'}{AD} = \frac{BB'}{BE} = \frac{CC'}{CF} = k \).

2. **Objective:**
   - Find all values of \( k \) such that \( \triangle A'B'C' \sim \triangle ABC \) for any non-equilateral triangle \( ABC \).

3. **Similarity Condition:**
   - For \( \triangle A'B'C' \sim \triangle ABC \), the corresponding angles must be equal, i.e., \( \angle A'B'C' = \angle ABC \), \( \angle B'C'A' = \angle BCA \), and \( \angle C'A'B' = \angle CAB \).

4. **Using Altitudes:**
   - Since \( A', B', C' \) are on the rays \( AD, BE, CF \) respectively, we can express their coordinates in terms of the coordinates of \( A, B, C \) and the altitudes \( AD, BE, CF \).

5. **Scaling Factor \( k \):**
   - Given \( \frac{AA'}{AD} = k \), we can write \( AA' = k \cdot AD \).
   - Similarly, \( BB' = k \cdot BE \) and \( CC' = k \cdot CF \).

6. **Coordinates of \( A', B', C' \):**
   - If \( D, E, F \) are the feet of the altitudes from \( A, B, C \) respectively, then:
     \[
     A' = A + k \cdot (D - A)
     \]
     \[
     B' = B + k \cdot (E - B)
     \]
     \[
     C' = C + k \cdot (F - C)
     \]

7. **Condition for Similarity:**
   - For \( \triangle A'B'C' \sim \triangle ABC \), the sides of \( \triangle A'B'C' \) must be proportional to the sides of \( \triangle ABC \) with the same ratio \( k \).

8. **Proportionality Check:**
   - The sides of \( \triangle A'B'C' \) are:
     \[
     A'B' = k \cdot AB, \quad B'C' = k \cdot BC, \quad C'A' = k \cdot CA
     \]
   - This implies that \( k \) must be such that the sides of \( \triangle A'B'C' \) are proportional to the sides of \( \triangle ABC \).

9. **Conclusion:**
   - For \( \triangle A'B'C' \sim \triangle ABC \), the ratio \( k \) must be the same for all sides, which is already given by the problem statement.
   - Therefore, \( k \) can be any positive real number.

The final answer is \( \boxed{ k } \) can be any positive real number.