Cyclic pentagon $ ABCDE$ has side lengths $ AB\equal{}BC\equal{}5$, $ CD\equal{}DE\equal{}12$, and $ AE \equal{} 14$. Determine the radius of its circumcircle.
To determine the radius of the circumcircle of the cyclic pentagon \(ABCDE\) with given side lengths \(AB = BC = 5\), \(CD = DE = 12\), and \(AE = 14\), we can use Ptolemy's Theorem and properties of cyclic polygons.

1. **Introduce a new point \(C'\)**:
   We introduce a point \(C'\) such that \(BC' = 12\) and \(C'D = 5\), on the minor arc \(BCD\). This helps us create symmetry and allows us to use Ptolemy's Theorem effectively.

2. **Apply Ptolemy's Theorem**:
   Ptolemy's Theorem states that for a cyclic quadrilateral \(ABCD\), the sum of the products of its two pairs of opposite sides is equal to the product of its diagonals:
   \[
   AB \cdot CD + BC \cdot AD = AC \cdot BD
   \]
   We can apply this theorem to various quadrilaterals within \(ABC'DE\).

3. **Form equations using Ptolemy's Theorem**:
   - For quadrilateral \(ABCE\):
     \[
     AB \cdot CE + BC \cdot AE = AC \cdot BE
     \]
     Substituting the given values:
     \[
     5 \cdot CE + 5 \cdot 14 = AC \cdot BE
     \]
     \[
     5CE + 70 = AC \cdot BE \quad \text{(1)}
     \]

   - For quadrilateral \(BC'DE\):
     \[
     BC' \cdot DE + C'D \cdot BE = BD \cdot C'E
     \]
     Substituting the given values:
     \[
     12 \cdot 12 + 5 \cdot BE = BD \cdot C'E
     \]
     \[
     144 + 5BE = BD \cdot C'E \quad \text{(2)}
     \]

   - For quadrilateral \(C'DAE\):
     \[
     C'D \cdot AE + DA \cdot C'E = DE \cdot AC'
     \]
     Substituting the given values:
     \[
     5 \cdot 14 + DA \cdot C'E = 12 \cdot AC'
     \]
     \[
     70 + DA \cdot C'E = 12AC' \quad \text{(3)}
     \]

4. **Solve the equations**:
   We now have three equations with three variables \(CE\), \(BE\), and \(C'E\). Solving these equations simultaneously will give us the values of these variables.

5. **Use the circumradius formula**:
   Once we have the values of the diagonals, we can use the formula for the circumradius \(R\) of a triangle with sides \(a\), \(b\), and \(c\) and area \(K\):
   \[
   R = \frac{abc}{4K}
   \]
   We need to find the area \(K\) of the triangle formed by the diagonals and one of the sides.

6. **Calculate the circumradius**:
   After finding the values of the diagonals and the area, we can substitute them into the formula to find the radius \(R\).

The final answer is \(\boxed{13}\).