Introduction:
In this problem, we are given a hyperbola with the equation x^2/9 - y^2/4 = 1. We need to find the locus of the middle points of the chords of this hyperbola that pass through the point (1, 2). The locus will be another hyperbola, and we need to determine its center.

Step 1: Find the equation of the chord:
Let's consider a general point (h, k) on the hyperbola. The equation of the chord passing through (h, k) and (1, 2) can be found using the slope-intercept form of a line. The slope of the chord is (k - 2)/(h - 1), and using the point-slope form, the equation of the chord is: y - 2 = (k - 2)/(h - 1)(x - 1).

Step 2: Find the midpoint of the chord:
The midpoint of the chord is given by the average of the x-coordinates and the average of the y-coordinates of the two endpoints. Let's denote the midpoint as (a, b). The x-coordinate of the midpoint is (h + 1)/2, and the y-coordinate is (k + 2)/2.

Step 3: Substitute the midpoint coordinates into the equation of the chord:
Substituting the midpoint coordinates (a, b) into the equation of the chord, we get: b - 2 = (k - 2)/(h - 1)(a - 1).

Step 4: Simplify and eliminate the parameter:
To eliminate the parameter, we can square both sides of the equation obtained in step 3 and simplify. This will give us an equation in terms of a and b only.

Step 5: Determine the center of the locus:
The equation obtained in step 4 represents the locus of the midpoints of the chords. By comparing the equation with the standard form of a hyperbola, we can determine the center of the locus.

Conclusion:
By following the steps outlined above, we can find the equation of the locus of the midpoints of the chords of the given hyperbola that pass through the point (1, 2). The locus will be another hyperbola, and its center can be determined by analyzing the simplified equation.