The equation of a circle passing through three given points can be determined using the concept of the perpendicular bisectors of the chords connecting these points.

Step 1: Find the midpoint of each chord
The midpoint of a chord can be found by taking the average of the coordinates of its endpoints. Let's find the midpoints of the chords formed by the given points.

Midpoint of chord passing through (1,2) and (2,1):
x-coordinate = (1 + 2)/2 = 1.5
y-coordinate = (2 + 1)/2 = 1.5
So, the midpoint of this chord is (1.5, 1.5).

Midpoint of chord passing through (2,1) and (3,5):
x-coordinate = (2 + 3)/2 = 2.5
y-coordinate = (1 + 5)/2 = 3
So, the midpoint of this chord is (2.5, 3).

Midpoint of chord passing through (3,5) and (1,2):
x-coordinate = (3 + 1)/2 = 2
y-coordinate = (5 + 2)/2 = 3.5
So, the midpoint of this chord is (2, 3.5).

Step 2: Find the slopes of the perpendicular bisectors
The slope of the perpendicular bisector of a chord is the negative reciprocal of the slope of the chord. Let's find the slopes of the perpendicular bisectors for the midpoints found in step 1.

Slope of the perpendicular bisector to the chord passing through (1,2) and (2,1):
Slope of the chord = (1.5 - 1.5)/(1.5 - 1.5) = undefined (vertical line)
Slope of the perpendicular bisector = -1/undefined = 0

Slope of the perpendicular bisector to the chord passing through (2,1) and (3,5):
Slope of the chord = (3 - 1)/(5 - 1) = 0.5
Slope of the perpendicular bisector = -1/0.5 = -2

Slope of the perpendicular bisector to the chord passing through (3,5) and (1,2):
Slope of the chord = (2.5 - 2)/(3.5 - 3) = 0.5
Slope of the perpendicular bisector = -1/0.5 = -2

Step 3: Find the equations of the perpendicular bisectors
Using the slopes found in step 2 and the midpoints from step 1, we can determine the equations of the perpendicular bisectors.

Equation of the perpendicular bisector to the chord passing through (1,2) and (2,1):
Using the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope:
y - 1.5 = 0(x - 1.5)
y - 1.5 = 0
y = 1.5

Equation of the perpendicular bisector to the chord passing through (2,1) and (3,5):
Using the point-s