The equation of the circle is **(x - 2)^2 + (y - 5)^2 = 10**. 

Here's how we find it:

1. **Find the slope of the tangent line:** The given line 2x - y + 1 = 0 has a slope of 2 (rewriting in slope-intercept form: y = 2x + 1).

2. **Find the slope of the radius:** The radius of the circle is perpendicular to the tangent line at the point of tangency. Therefore, the slope of the radius is the negative reciprocal of 2, which is -1/2.

3. **Find the equation of the radius:** We know the radius passes through (2, 5) and has a slope of -1/2. Using the point-slope form, we get:

  y - 5 = (-1/2)(x - 2)

4. **Find the center of the circle:** The center of the circle lies on the line x + y = 9. Substituting the equation of the radius (from step 3) into this equation, we get:

  x + (-1/2)(x - 2) + 5 = 9

  Solving for x, we get x = 4. Substituting this back into x + y = 9, we get y = 5. Therefore, the center of the circle is (4, 5).

5. **Find the radius of the circle:** The distance between the center (4, 5) and the point of tangency (2, 5) is the radius. This distance is 2.

6. **Write the equation of the circle:** Using the standard form of the circle equation, we get:

  (x - 4)^2 + (y - 5)^2 = 2^2

  **(x - 2)^2 + (y - 5)^2 = 10**