How to Find the Center of a Circle When Given Three Points

To find the center of a circle given three points that lie on its circumference, you can use the following step-by-step geometric method:

1. Label the Points: Let’s call the three points A, B, and C.
2. Find the Midpoints: Calculate the midpoints of the line segments AB and BC. The midpoint M1 of segment AB can be found using the formula:

   M1 = ((xA + xB) / 2, (yA + yB) / 2)

3. Determine the Slopes: Next, calculate the slopes of lines AB and BC. The slope of a line between two points can be calculated as:

   m = (y2 – y1) / (x2 – x1)

4. Find the Perpendicular Slopes: The slopes of the perpendicular bisectors will be the negative reciprocals of the slopes of the original lines. For example, if slope of AB is m1, then the slope of its perpendicular bisector is -1/m1.
5. Equation of Perpendicular Bisectors: Use the midpoints and the perpendicular slopes to write the equations of the two perpendicular bisectors. For M1 with slope m, the equation will look like:

   (y – yM1) = m(x – xM1)

6. Find the Intersection: Finally, solve the system of equations formed by the two perpendicular bisectors. The intersection point is the center of the circle.

This method relies on basic properties of geometry: the perpendicular bisector of a chord goes through the center of a circle. By finding the intersection of two such bisectors, you can accurately locate the center of the circle defined by the three points.