Given is triangle $ABC$ with incenter $I$ and $A$-excenter $J$. Circle $\omega_b$ centered at point $O_b$ passes through point $B$ and is tangent to line $CI$ at point $I$. Circle $\omega_c$ with center $O_c$ passes through point $C$ and touches line $BI$ at point $I$. Let $O_bO_c$ and $IJ$ intersect at point $K$. Find the ratio $IK/KJ$.
1. **Identify Key Points and Properties:**
   - Let \( I \) be the incenter of \(\triangle ABC\).
   - Let \( J \) be the \( A \)-excenter of \(\triangle ABC\).
   - Circle \(\omega_b\) is centered at \( O_b \), passes through \( B \), and is tangent to line \( CI \) at \( I \).
   - Circle \(\omega_c\) is centered at \( O_c \), passes through \( C \), and is tangent to line \( BI \) at \( I \).
   - \( O_bO_c \) and \( IJ \) intersect at point \( K \).

2. **Midpoint and Perpendicular Bisector:**
   - Let \( M \) be the midpoint of \( IJ \).
   - Since \( M \) is the midpoint, \( MI = MJ \).

3. **Perpendicular Bisector and Parallel Lines:**
   - \( MO_b \) is the perpendicular bisector of \( BI \) because \( \omega_b \) is tangent to \( CI \) at \( I \).
   - Similarly, \( MO_c \) is the perpendicular bisector of \( CI \) because \( \omega_c \) is tangent to \( BI \) at \( I \).

4. **Parallelogram Formation:**
   - Since \( MO_b \) and \( MO_c \) are perpendicular bisectors, \( O_bI \parallel MO_c \) and \( O_cI \parallel MO_b \).
   - Therefore, \( IO_bMO_c \) forms a parallelogram.

5. **Intersection and Ratio Calculation:**
   - In a parallelogram, the diagonals bisect each other.
   - Since \( M \) is the midpoint of \( IJ \) and \( IO_bMO_c \) is a parallelogram, \( K \) (the intersection of \( O_bO_c \) and \( IJ \)) must be the midpoint of \( IJ \).

6. **Ratio \( IK/KJ \):**
   - Since \( K \) is the midpoint of \( IJ \), \( IK = KJ \).
   - Therefore, the ratio \( \frac{IK}{KJ} = \frac{1}{1} = 1 \).

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