Determining the Center and Radius of a Circle Passing Through Three Points

Given three points on a circle, we can determine the circle's center and radius using geometric properties and formulas. This article will guide you through a detailed step-by-step process using the midpoint and slope methods, as well as a direct construction method.

Introduction to Circle Properties

A circle in a plane is a set of points that are equidistant from a fixed point, known as the center. The distance from the center to any point on the circle is called the radius. Given three points on a circle, we can use various methods to determine the circle's center and radius.

Method 1: Using Midpoints and Slopes

Let's consider three points on a circle, (A(x_1, y_1)), (B(x_2, y_2)), and (C(x_3, y_3)). The center of the circle, (M(x_4, y_4)), can be found by finding the intersection of the perpendicular bisectors of two chords (AB) and (BC).

Step 1: Find Midpoints of Chords

The midpoint of (AB) is (P left( frac{x_1 x_2}{2}, frac{y_1 y_2}{2} right)), and the midpoint of (BC) is (Q left( frac{x_2 x_3}{2}, frac{y_2 y_3}{2} right)).

Step 2: Slopes of Chords

The slope of (AB) is (frac{y_2 - y_1}{x_2 - x_1}), and the slope of (BC) is (frac{y_3 - y_2}{x_3 - x_2}).

Step 3: Slopes of Perpendicular Bisectors

The slope of the line perpendicular to (AB) is -frac{x_2 - x_1}{y_2 - y_1}), and the slope of the line perpendicular to (BC) is -frac{x_3 - x_2}{y_3 - y_2}.

Step 4: Equations of Perpendicular Bisectors

Using the point-slope form, the equation of line (PR) through (P) is (y - frac{y_1 y_2}{2} -frac{x_2 - x_1}{y_2 - y_1} left(x - frac{x_1 x_2}{2}right)), and the equation of line (QS) through (Q) is (y - frac{y_2 y_3}{2} -frac{x_3 - x_2}{y_3 - y_2} left(x - frac{x_2 x_3}{2}right)).

Step 5: Find the Intersection Point

Solve the two equations to find the intersection point, which is the center (M(x_4, y_4)) of the circle.

Step 6: Calculate the Radius

The radius of the circle is the distance from the center (M(x_4, y_4)) to any of the points (A), (B), or (C), given by (sqrt{(x_1 - x_4)^2 (y_1 - y_4)^2}).

Method 2: Direct Construction

Another way to find the center and radius is by using the properties of the perpendicular bisectors directly.

Step 1: Draw the Perpendicular Bisectors

For points (A), (B), and (C), draw the perpendicular bisectors of two chords, say (AB) and (BC).

Step 2: Intersection Point

The intersection of these two perpendicular bisectors is the center of the circle, (M(x_4, y_4)).

Step 3: Calculate the Radius

The radius of the circle is the distance from the center (M(x_4, y_4)) to any of the points (A), (B), or (C), given by (sqrt{(x_1 - x_4)^2 (y_1 - y_4)^2}).

Conclusion

Using either method, we can effectively determine the center and radius of a circle that passes through three given points. Whether you prefer the algebraic approach or the geometric construction, understanding these methods enhances your problem-solving skills in geometry and has practical applications in various fields, including engineering and architecture.