To solve the problem, we follow these steps:

1. Find the midpoint \(M\) of \(AB\):

\(M = \left( \frac{2 + 8}{2}, \frac{6 + 10}{2} \right) = (5, 8)\)

2. Calculate the gradient of \(AB\):

\(\text{Gradient of } AB = \frac{10 - 6}{8 - 2} = \frac{2}{3}\)

3. Find the gradient of the perpendicular bisector:

\(\text{Perpendicular gradient} = -\frac{3}{2}\)

4. Use the point-slope form to find the equation of the perpendicular bisector:

\(y - 8 = -\frac{3}{2}(x - 5)\)

Expanding and rearranging gives:

\(2y + 3x = 31\)

5. Find the equation of line \(BC\):

\(y = 5(x - 6) \Rightarrow y = 5x - 30\)

6. Solve the system of equations:

\(2y + 3x = 31\)

\(y = 5x - 30\)

Substitute \(y = 5x - 30\) into \(2y + 3x = 31\):

\(2(5x - 30) + 3x = 31\)

\(10x - 60 + 3x = 31\)

\(13x = 91\)

\(x = 7\)

Substitute \(x = 7\) back into \(y = 5x - 30\):

\(y = 5(7) - 30 = 5\)

Thus, the coordinates of \(D\) are \((7, 5)\).