In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}$.  Then $\sin \left( \angle DAE \right)$ can be expressed in the form $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime.  Find $a+b+c$.
1. **Define the problem and setup the triangle:**
   - Consider an equilateral triangle \( \triangle ABC \) with side length \( 6 \).
   - Points \( D \) and \( E \) trisect \( \overline{BC} \), so \( BD = DE = EC = 2 \).

2. **Determine the coordinates of points \( D \) and \( E \):**
   - Place \( A \) at the origin \((0, 0)\), \( B \) at \((3, 3\sqrt{3})\), and \( C \) at \((6, 0)\).
   - Since \( D \) and \( E \) trisect \( \overline{BC} \), their coordinates are:
     - \( D = (4, 2\sqrt{3}) \)
     - \( E = (5, \sqrt{3}) \)

3. **Calculate the lengths of perpendiculars from \( D \) and \( E \) to \( \overline{BC} \):**
   - Drop perpendiculars from \( D \) and \( E \) to \( \overline{BC} \), meeting \( BC \) at points \( F \) and \( G \) respectively.
   - By properties of \( 30^\circ-60^\circ-90^\circ \) triangles:
     - \( DF = 2\sqrt{3} \)
     - \( EG = \sqrt{3} \)

4. **Calculate the distances \( AF \) and \( AG \):**
   - \( F \) and \( G \) are the feet of the perpendiculars from \( D \) and \( E \) respectively.
   - \( F \) and \( G \) are on \( \overline{BC} \) such that:
     - \( FG = 1 \)
     - \( AF = 4 \)
     - \( AG = 5 \)

5. **Calculate the lengths \( AD \) and \( AE \):**
   - Using the Pythagorean theorem:
     \[
     AD = \sqrt{AF^2 + DF^2} = \sqrt{4^2 + (2\sqrt{3})^2} = \sqrt{16 + 12} = \sqrt{28} = 2\sqrt{7}
     \]
     \[
     AE = \sqrt{AG^2 + EG^2} = \sqrt{5^2 + (\sqrt{3})^2} = \sqrt{25 + 3} = \sqrt{28} = 2\sqrt{7}
     \]

6. **Calculate the trigonometric values:**
   - \(\sin(\angle DAF) = \frac{DF}{AD} = \frac{2\sqrt{3}}{2\sqrt{7}} = \frac{\sqrt{3}}{\sqrt{7}}\)
   - \(\cos(\angle DAF) = \frac{AF}{AD} = \frac{4}{2\sqrt{7}} = \frac{2}{\sqrt{7}}\)
   - \(\sin(\angle EAG) = \frac{EG}{AE} = \frac{\sqrt{3}}{2\sqrt{7}}\)
   - \(\cos(\angle EAG) = \frac{AG}{AE} = \frac{5}{2\sqrt{7}}\)

7. **Use the angle subtraction formula for sine:**
   \[
   \sin(\angle DAE) = \sin(\angle DAF - \angle EAG) = \sin(\angle DAF)\cos(\angle EAG) - \cos(\angle DAF)\sin(\angle EAG)
   \]
   \[
   \sin(\angle DAE) = \left(\frac{\sqrt{3}}{\sqrt{7}}\right)\left(\frac{5}{2\sqrt{7}}\right) - \left(\frac{2}{\sqrt{7}}\right)\left(\frac{\sqrt{3}}{2\sqrt{7}}\right)
   \]
   \[
   \sin(\angle DAE) = \frac{5\sqrt{3}}{2 \cdot 7} - \frac{2\sqrt{3}}{2 \cdot 7} = \frac{5\sqrt{3} - 2\sqrt{3}}{14} = \frac{3\sqrt{3}}{14}
   \]

8. **Identify the values of \( a \), \( b \), and \( c \):**
   - \( a = 3 \)
   - \( b = 3 \)
   - \( c = 14 \)

9. **Sum the values:**
   \[
   a + b + c = 3 + 3 + 14 = 20
   \]

The final answer is \(\boxed{20}\)