Introduction:

To find the equation of a circle that is circumscribed about a triangle, we can make use of the fact that the center of the circle will be the point of intersection of the perpendicular bisectors of the sides of the triangle.

Step-by-step Solution:

1. Find the midpoints of the sides:
- The midpoint of the line segment joining (-2, 3) and (5, 2) can be found using the midpoint formula:
Midpoint of AB = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint of AB = ((-2 + 5)/2, (3 + 2)/2)
Midpoint of AB = (3/2, 5/2)
- Similarly, the midpoints of the line segments joining the other two pairs of vertices can be found:
Midpoint of BC = ((5 + 6)/2, (2 - 1)/2)
Midpoint of BC = (11/2, 1/2)
Midpoint of AC = ((-2 + 6)/2, (3 - 1)/2)
Midpoint of AC = (2, 1)

2. Find the slopes of the sides:
- The slope of the line joining (-2, 3) and (5, 2) can be found using the slope formula:
Slope of AB = (y2 - y1)/(x2 - x1)
Slope of AB = (2 - 3)/(5 - (-2))
Slope of AB = -1/7
- Similarly, the slopes of the other two sides can be found:
Slope of BC = (1/2 - 2)/(11/2 - 6)
Slope of BC = -1/7
Slope of AC = (1 - 3)/(2 - (-2))
Slope of AC = 1/2

3. Find the equations of the perpendicular bisectors:
- The equation of the perpendicular bisector of AB can be found using the point-slope form:
y - y1 = m⊥(x - x1)
Since the midpoint of AB is (3/2, 5/2) and the slope of AB is -1/7, the equation of the perpendicular bisector of AB is:
y - 5/2 = 7/1(x - 3/2)
Simplifying, we get:
y - 5/2 = 7x - 21/2
y = 7x - 21/2 + 5/2
y = 7x - 8/2
y = 7x - 4

- Similarly, the equations of the perpendicular bisectors of BC and AC can be found:
Perpendicular bisector of BC: y - 1/2 = 7/1(x - 11/2)
Simplifying, we get:
y - 1/2 = 7x - 77/2
y = 7x - 77/2 + 1/2
y = 7x - 76/2
y = 7x