Equation of circle when three points on the circle are given

• Difficulty Level : Expert
• Last Updated : 19 Jul, 2021

Given three coordinates that lie on a circle, (x1, y1), (x2, y2), and (x3, y3). The task is to find the equation of the circle and then print the centre and the radius of the circle.
Equation of circle in general form is x² + y² + 2gx + 2fy + c = 0 and in radius form is (x – h)² + (y -k)² = r², where (h, k) is the centre of the circle and r is the radius.
Examples:

Input: x1 = 1, y1 = 0, x2 = -1, y2 = 0, x3 = 0, y3 = 1
Output:
Centre = (0, 0)
The equation of the circle is x2 + y2 = 1.
Input: x1 = 1, y1 = -6, x2 = 2, y2 = 1, x3 = 5, y3 = 2
Output:
Centre = (5, -3)
Equation of the circle is x2 + y2 -10x + 6y + 9 = 0

Approach: As we know all three-point lie on the circle, so they will satisfy the equation of a circle and by putting them in the general equation we get three equations with three variables g, f, and c, and by further solving we can get the values. We can derive the formula to obtain the value of g, f, and c as:

Putting coordinates in eqn of circle, we get:
x12 + y12 + 2gx1 + 2fy1 + c = 0 – (1)
x22 + y22 + 2gx2 + 2fy2 + c = 0 – (2)
x32 + y32 + 2gx3 + 2fy3 + c = 0 – (3)
From (1) we get, 2gx1 = -x12 – y12 – 2fy1 – c – (4)
From (1) we get, c = -x12 – y12 – 2gx1 – 2fy1 – (5)
From (3) we get, 2fy3 = -x32 – y32 – 2gx3 – c – (6)
Subtracting eqn (2) from eqn (1) we get
2g( x1 – x2 ) = ( x22 -x12 ) + ( y22 – y12 ) + 2f( y2 – y1 ) – (A)
Now putting eqn (5) in (6) we get
2fy3 = -x32 – y32 – 2gx3 + x12 + y12 + 2gx1 + 2fy1 – (7)
Now putting value of 2g from eqn (A) in (7) we get
2f = ( ( x12 – x32 )( x1 – x2 ) +( y12 – y32 )( x1 – x2 ) + ( x22 – x12 )( x1 – x3 ) + ( y22 – y12 )( x1 – x3 ) ) / ( y3 – y1 )( x1 – x2 ) – ( y2 – y1 )( x1 – x3 )
Similarly we can obtain the values of 2g :
2g = ( ( x12 – x32 )( y1 – x2 ) +( y12 – y32 )( y1 – y2 ) + ( x22 – x12 )( y1 – y3) + ( y22 – y12 )( y1 – y3 ) ) / ( x3 -x1 )( y1 – y2 ) – ( x2 – x1 )( y1 – y3 )
Putting 2g and 2f in eqn (5) we get the value of c and know we had the equation of circle as x2 + y2 + 2gx + 2fy + c = 0

Below is the implementation of the above approach:

C++

 // C++ implementation of the approach#include using namespace std; // Function to find the circle on// which the given three points lievoid findCircle(int x1, int y1, int x2, int y2, int x3, int y3){    int x12 = x1 - x2;    int x13 = x1 - x3;     int y12 = y1 - y2;    int y13 = y1 - y3;     int y31 = y3 - y1;    int y21 = y2 - y1;     int x31 = x3 - x1;    int x21 = x2 - x1;     // x1^2 - x3^2    int sx13 = pow(x1, 2) - pow(x3, 2);     // y1^2 - y3^2    int sy13 = pow(y1, 2) - pow(y3, 2);     int sx21 = pow(x2, 2) - pow(x1, 2);    int sy21 = pow(y2, 2) - pow(y1, 2);     int f = ((sx13) * (x12)             + (sy13) * (x12)             + (sx21) * (x13)             + (sy21) * (x13))            / (2 * ((y31) * (x12) - (y21) * (x13)));    int g = ((sx13) * (y12)             + (sy13) * (y12)             + (sx21) * (y13)             + (sy21) * (y13))            / (2 * ((x31) * (y12) - (x21) * (y13)));     int c = -pow(x1, 2) - pow(y1, 2) - 2 * g * x1 - 2 * f * y1;     // eqn of circle be x^2 + y^2 + 2*g*x + 2*f*y + c = 0    // where centre is (h = -g, k = -f) and radius r    // as r^2 = h^2 + k^2 - c    int h = -g;    int k = -f;    int sqr_of_r = h * h + k * k - c;     // r is the radius    float r = sqrt(sqr_of_r);     cout << "Centre = (" << h << ", " << k << ")" << endl;    cout << "Radius = " << r;} // Driver codeint main(){    int x1 = 1, y1 = 1;    int x2 = 2, y2 = 4;    int x3 = 5, y3 = 3;    findCircle(x1, y1, x2, y2, x3, y3);     return 0;}

Java

 // Java implementation of the approachimport java.text.*; class GFG{     // Function to find the circle on// which the given three points liestatic void findCircle(int x1, int y1,                        int x2, int y2,                        int x3, int y3){    int x12 = x1 - x2;    int x13 = x1 - x3;     int y12 = y1 - y2;    int y13 = y1 - y3;     int y31 = y3 - y1;    int y21 = y2 - y1;     int x31 = x3 - x1;    int x21 = x2 - x1;     // x1^2 - x3^2    int sx13 = (int)(Math.pow(x1, 2) -                    Math.pow(x3, 2));     // y1^2 - y3^2    int sy13 = (int)(Math.pow(y1, 2) -                    Math.pow(y3, 2));     int sx21 = (int)(Math.pow(x2, 2) -                    Math.pow(x1, 2));                         int sy21 = (int)(Math.pow(y2, 2) -                    Math.pow(y1, 2));     int f = ((sx13) * (x12)            + (sy13) * (x12)            + (sx21) * (x13)            + (sy21) * (x13))            / (2 * ((y31) * (x12) - (y21) * (x13)));    int g = ((sx13) * (y12)            + (sy13) * (y12)            + (sx21) * (y13)            + (sy21) * (y13))            / (2 * ((x31) * (y12) - (x21) * (y13)));     int c = -(int)Math.pow(x1, 2) - (int)Math.pow(y1, 2) -                                2 * g * x1 - 2 * f * y1;     // eqn of circle be x^2 + y^2 + 2*g*x + 2*f*y + c = 0    // where centre is (h = -g, k = -f) and radius r    // as r^2 = h^2 + k^2 - c    int h = -g;    int k = -f;    int sqr_of_r = h * h + k * k - c;     // r is the radius    double r = Math.sqrt(sqr_of_r);    DecimalFormat df = new DecimalFormat("#.#####");    System.out.println("Centre = (" + h + "," + k + ")");    System.out.println("Radius = " + df.format(r));} // Driver codepublic static void main (String[] args){    int x1 = 1, y1 = 1;    int x2 = 2, y2 = 4;    int x3 = 5, y3 = 3;    findCircle(x1, y1, x2, y2, x3, y3);}} // This code is contributed by chandan_jnu

Python3

 # Python3 implementation of the approachfrom math import sqrt # Function to find the circle on# which the given three points liedef findCircle(x1, y1, x2, y2, x3, y3) :    x12 = x1 - x2;    x13 = x1 - x3;     y12 = y1 - y2;    y13 = y1 - y3;     y31 = y3 - y1;    y21 = y2 - y1;     x31 = x3 - x1;    x21 = x2 - x1;     # x1^2 - x3^2    sx13 = pow(x1, 2) - pow(x3, 2);     # y1^2 - y3^2    sy13 = pow(y1, 2) - pow(y3, 2);     sx21 = pow(x2, 2) - pow(x1, 2);    sy21 = pow(y2, 2) - pow(y1, 2);     f = (((sx13) * (x12) + (sy13) *          (x12) + (sx21) * (x13) +          (sy21) * (x13)) // (2 *          ((y31) * (x12) - (y21) * (x13))));                 g = (((sx13) * (y12) + (sy13) * (y12) +          (sx21) * (y13) + (sy21) * (y13)) //          (2 * ((x31) * (y12) - (x21) * (y13))));     c = (-pow(x1, 2) - pow(y1, 2) -         2 * g * x1 - 2 * f * y1);     # eqn of circle be x^2 + y^2 + 2*g*x + 2*f*y + c = 0    # where centre is (h = -g, k = -f) and    # radius r as r^2 = h^2 + k^2 - c    h = -g;    k = -f;    sqr_of_r = h * h + k * k - c;     # r is the radius    r = round(sqrt(sqr_of_r), 5);     print("Centre = (", h, ", ", k, ")");    print("Radius = ", r); # Driver codeif __name__ == "__main__" :         x1 = 1 ; y1 = 1;    x2 = 2 ; y2 = 4;    x3 = 5 ; y3 = 3;    findCircle(x1, y1, x2, y2, x3, y3); # This code is contributed by Ryuga

C#

 // C# implementation of the approachusing System; class GFG{     // Function to find the circle on// which the given three points liestatic void findCircle(int x1, int y1,                        int x2, int y2,                        int x3, int y3){    int x12 = x1 - x2;    int x13 = x1 - x3;     int y12 = y1 - y2;    int y13 = y1 - y3;     int y31 = y3 - y1;    int y21 = y2 - y1;     int x31 = x3 - x1;    int x21 = x2 - x1;     // x1^2 - x3^2    int sx13 = (int)(Math.Pow(x1, 2) -                    Math.Pow(x3, 2));     // y1^2 - y3^2    int sy13 = (int)(Math.Pow(y1, 2) -                    Math.Pow(y3, 2));     int sx21 = (int)(Math.Pow(x2, 2) -                    Math.Pow(x1, 2));                         int sy21 = (int)(Math.Pow(y2, 2) -                    Math.Pow(y1, 2));     int f = ((sx13) * (x12)            + (sy13) * (x12)            + (sx21) * (x13)            + (sy21) * (x13))            / (2 * ((y31) * (x12) - (y21) * (x13)));    int g = ((sx13) * (y12)            + (sy13) * (y12)            + (sx21) * (y13)            + (sy21) * (y13))            / (2 * ((x31) * (y12) - (x21) * (y13)));     int c = -(int)Math.Pow(x1, 2) - (int)Math.Pow(y1, 2) -                                2 * g * x1 - 2 * f * y1;     // eqn of circle be x^2 + y^2 + 2*g*x + 2*f*y + c = 0    // where centre is (h = -g, k = -f) and radius r    // as r^2 = h^2 + k^2 - c    int h = -g;    int k = -f;    int sqr_of_r = h * h + k * k - c;     // r is the radius    double r = Math.Round(Math.Sqrt(sqr_of_r), 5);     Console.WriteLine("Centre = (" + h + "," + k + ")");    Console.WriteLine("Radius = " + r);} // Driver codestatic void Main(){    int x1 = 1, y1 = 1;    int x2 = 2, y2 = 4;    int x3 = 5, y3 = 3;    findCircle(x1, y1, x2, y2, x3, y3);}} // This code is contributed by chandan_jnu


Output:
Centre = (3, 2)