# Explainer: Equation of a Circle Passing through Three Noncollinear Points

In this explainer, we will learn how to find the equation of a circle passing through three noncollinear points that form a right triangle.

You should know the two forms of the general equation of a circle.

### Equation of a Circle in Radius-Center Form

The equation of a circle centered at and of radius is

### Equation of a Circle in General Form

The equation of a circle can be written in the form

### How to Write the Equation of a Circle That Passes through Three Points in the Form (𝑥 − ℎ)² + (𝑦 − 𝑘)² = 𝑟²

1. The coordinates of the three given points on the circle must satisfy the equation of the circle. With , , and the three given points, we can write
Note that given , the above equations are simply saying that the distance between the center of the circle, of coordinates , and each of the three points is constant and equal to the radius.
2. Since the three equations are in the form Expression , they can be rearranged in a system of two equations with two unknowns by writing and . (Note that any two combinations of the three equations work.) We get
By expanding the brackets, we see that the terms and cancel out:
By solving this system of equations, we find the coordinates of the circle center .
3. The last stage is to plug in these values of and in one of our first three equations to find the value of .
4. The equation of the circle is then with the values of , and we have found.

Let’s see how this method is implemented when we have the coordinates of the three points.

### Example 1: Writing the Equation of a Circle That Passes through Three Points in Center-Radius Form

Find the equation of the circle that passes through the points , , and .

1. We are looking for the equation of the circle in the form . Let’s write that the coordinates of the three points satisfy the equation of the circle:
2. We arrange this system of three equations in a system of two equations:
By expanding the brackets, we find
By rearranging and collecting like terms, we get
From the second equation, we get . By plugging in this value into the first equation, we find . And by plugging this value of in one of the two equations above, we find .
3. By plugging in these values of and in one of the first three equations, for instance, , we have and thus
4. The equation of the circle is: .

Let’s see now another method to find the equation of a circle that passes through three points when we want to have the equation in the form .

### How to Write the Equation of a Circle That Passes through Three Points in the Form 𝑥² + 𝑦² + 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0

1. The coordinates of the three given points on the circle must satisfy the equation of the circle. With , , and the three given points, we can write
2. Solve the system of three equations to find the unknowns , , and .
3. Plug the values found in the general equation .

### Example 2: Writing the Equation of a Circle That Passes through Three Points in General Form

Determine the general equation of the shown circle passing through the origin point and the two points and .