You are given three points in a plane. Construct an algorithm to determine the center and radius of the unique circle that passes through all three points.?

To determine the center and radius of the unique circle that passes through three non-collinear points (A(x_A, y_A), B(x_B, y_B), and C(x_C, y_C), you may follow these steps:

1. Find the midpoints of the line segments AB and BC, denoted as D and E, respectively.

$$x_D = \frac{x_A + x_B}{2},\ y_D = \frac{y_A + y_B}{2}$$

$$x_E = \frac{x_B + x_C}{2},\ y_E = \frac{y_B + y_C}{2}$$

2. Calculate the slopes of the lines AB and BC, denoted as m_AB and m_BC, respectively.

$$m_{AB} = \frac{y_B - y_A}{x_B - x_A}$$

$$m_{BC} = \frac{y_C - y_B}{x_C - x_B}$$

3. Determine the slopes of the perpendicular bisectors of line segments AB and BC, denoted as m_D and m_E, respectively.

$$m_D = -\frac{1}{m_{AB}} = -\frac{x_B-x_A}{y_B-y_A}$$

$$m_E = -\frac{1}{m_{BC}} = -\frac{x_C-x_B}{y_C-y_B}$$

4. For the perpendicular bisectors of line segments AB and BC, calculate the equations of the lines containing the bisectors of both line segments. Using the point-slope form, these are:

y − y_D = m_D(x − x_D)

y − y_E = m_E(x − x_E)

5. Next, solve the system of equations in step 4 to find the coordinates of the center O(x_O, y_O) of the circle.

6. Finally, compute the radius of the circle by finding the distance from the center to any of the original points, say point A. To calculate the distance between two points, you may use the distance formula:

$$r = \sqrt {(x_O - x_A)^2 + (y_O - y_A)^2}$$

Once you have the center and radius, you’ve found the unique circle that passes through the three points.

For an algorithm, you may implement these steps in any programming language or mathematical tool, making sure to handle special cases such as vertical lines, horizontal lines, and duplicate points.