First, we will need to determine the *slopes of two lines - lines AB and BC*.
Slope Line AB = (y1 -y2) ÷ (x1 -x2) = (2 --4) ÷ (9 -3) = (6) ÷ (6) = 1

Slope Line BC = (y3 -y2) ÷ (x3 -x2) = (-6 --4) ÷ (5 -3) = (-2) ÷ (2) = -1

Now, we need to find the *'x' coordinate* of the circle's center which is:

xctr = [slope AB * slope BC * (Y3 -Y1) + slope AB * (X2 +X3) -slope BC * (X1 +X2)] ÷ [2 * (slope AB -slope BC)]

xctr = [ (1 * -1 * (-6 -2)) + (1 * (3 + 5)) - (-1 * (9 + 3))] ÷ 2 * (1 --1)

xctr = (-1 * (-8)) + 8 -(-12) ÷ 4

xctr = (8 + 8 + 12) ÷ 4

xctr = 7

To find the *'y' coordinate* of the circle's center we use this formula:

yctr = -(1/slope AB)*(xctr-[(x1 +x2)/2)] + (y1 +y2)/2

yctr = -(1/1) * (7 -[(9 +3)/2]) + (2 -4)/2

yctr = (-1 * (7 -6)) -(2/2)

yctr = (-1 * 1) -1

yctr = -2

Circle's Center is located at: (7, -2)

Finally, to calculate the *circle's radius*, we use this formula:

radius = Square Root [(x1 -xCtr)^2 + (y1 -yCtr)^2)]

where (x1, y1) can be *any* of the three points but let's use (9, 2)

radius = Square Root [(9 -7)^2 + (2 --2)^2)]

radius = Square Root [(2)^2 + (4)^2)]

radius = Square Root (20)

radius = 4.472135955

To calculate the *circle's equation*, insert those three numbers into this equation.