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Descartes’ Circle Theorem with implementation

  • Last Updated : 23 Jul, 2021

In geometry, Descartes’ theorem states that for every four mutually tangent circles, the radii of the circles satisfy a certain quadratic equation. One can construct a fourth circle tangent to three given, mutually tangent circles. 
The Descartes’ Circle Theorem helps us to find the radius of a circle when there are 4 circles with positive integer radius r1, r2, r3 and r4 as shown in the figure below. It finds the radius r4 of the circle formed by three circles of radius r1, r2, r3 as shown in the image below. 
(Note that the circles in the picture below are tangent to each other.)
 

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Examples: 
 



Input: r1 = 1, r2 = 1, r3 = 1 
Output: 0.154701
Input: r1 =23, r2 = 46, r3 = 69 
Output: 6.000000

 

 

The theorem says that the reciprocals of radii, or “curvatures”, of these circles 
 

k_1 = 1/r_1, k_2 = 1/r_2, k_3 = 1/r_3\:and\:k_4 = 1/r_4
satisfies the following relation:
2(k_1^2+k_2^2+k_3^2+k_4^2) = (k_1+k_2+k_3+k_4)^2

One of the circles can be enclosing among all the remaining three, as in the figure, in which case the corresponding curvature (k_4 = -1/r_4)          here is considered negative and the above relation still holds.
If k_1, k_2, k_3          are known, one can solve for k4, 
k_4 = k_1 + k_2 + k_3 \pm 2\sqrt{k_1 k_2 + k_2 k_3 + k_1 k_3}
On solving the above equation we get the radius of the fourth circle. The formula for finding the fourth circle’s radius is: 
 

r_4 = (r_1*r_2*r_3)\, /\, (r_1*r_2 + r_2*r_3 + r_3*r_1 + 2.0\, *\, \sqrt{(r_1*r_2*r_3*(r_1+r_2+r_3))})

Thus if r1, r2, and r3 are known, r4 can easily be calculated using the above formula. 
Below is the implementation of the above approach: 
 

CPP




// C++ implementation of the
// above formulae
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the fourth circle's
// when three radius are given
double findRadius(double r1, double r2, double r3)
{
    double r4 = (r1 * r2 * r3)
                / (r1 * r2 + r2 * r3
                   + r1 * r3 + 2.0 * sqrt(r1 * r2 * r3 * (r1 + r2 + r3)));
 
    return r4;
}
 
// Driver code
int main()
{
    // Radius of three circles
    double r1 = 1;
    double r2 = 1;
    double r3 = 1;
 
    // Calculation of r4 using formula given above
    double r4 = findRadius(r1, r2, r3);
 
    cout << "The radius of fourth circle: " << r4;
    return 0;
}

Java




/*package whatever //do not write package name here */
// Java implementation of the
// above formulae
import java.io.*;
 
class GFG
{
   
        // Function to find the fourth circle's
        // when three radius are given
        static double findRadius(double r1, double r2, double r3)
        {
        double r4 = (r1 * r2 * r3)
                / (r1 * r2 + r2 * r3
                   + r1 * r3 + 2.0 * Math.sqrt(r1 * r2 * r3 * (r1 + r2 + r3)));
 
        return r4;
        }
   
      // Driver code
    public static void main (String[] args)
    {
       
        // Radius of three circles
    double r1 = 1;
    double r2 = 1;
    double r3 = 1;
 
    // Calculation of r4 using formula given above
    double r4 = findRadius(r1, r2, r3);
        System.out.println("The radius of fourth circle: " + r4);
    }
}
 
// This code is contributed by CoderSaty.

Python3




# Python 3 implementation of the
# above formulae
 
from math import sqrt
 
# Function to find the fourth circle's
# when three radius are given
def findRadius(r1, r2, r3):
    r4 = (r1 * r2 * r3) / (r1 * r2 + r2 * r3 + r1 * r3 + 2.0 * sqrt(r1 * r2 * r3 * (r1 + r2 + r3)))
 
    return r4
 
# Driver code
if __name__ == '__main__':
   
    # Radius of three circles
    r1 = 1
    r2 = 1
    r3 = 1
 
    # Calculation of r4 using formula given above
    r4 = findRadius(r1, r2, r3)
 
    print("The radius of fourth circle:",r4)
     
    # This code is contributed by SURENDRA_GANGWAR.

C#




// C# implementation of the
// above formulae
using System;
 
class GFG
{
 
  // Function to find the fourth circle's
  // when three radius are given
  static double findRadius(double r1, double r2, double r3)
  {
    double r4 = (r1 * r2 * r3)
      / (r1 * r2 + r2 * r3
         + r1 * r3 + 2.0 * Math.Sqrt(r1 * r2 * r3 * (r1 + r2 + r3)));
 
    return r4;
  }
 
  // Driver code
  public static void Main (String[] args)
  {
 
    // Radius of three circles
    double r1 = 1;
    double r2 = 1;
    double r3 = 1;
 
    // Calculation of r4 using formula given above
    double r4 = Math.Round(findRadius(r1, r2, r3),6);
 
    Console.Write("The radius of fourth circle: " + r4);
  }
}
 
// This code is contributed by shivanisinghss2110

Javascript




<script>
// Javascript implementation of the
// above formulae
 
// Function to find the fourth circle's
// when three radius are given
function findRadius(r1, r2, r3) {
  let r4 =
    (r1 * r2 * r3) /
    (r1 * r2 +
      r2 * r3 +
      r1 * r3 +
      2.0 * Math.sqrt(r1 * r2 * r3 * (r1 + r2 + r3)));
 
  return r4.toFixed(6);
}
 
// Driver code
 
// Radius of three circles
let r1 = 1;
let r2 = 1;
let r3 = 1;
 
// Calculation of r4 using formula given above
let r4 = findRadius(r1, r2, r3);
 
document.write("The radius of fourth circle: " + r4);
 
// This code is contributed by gfgking.
</script>
Output: 
The radius of fourth circle: 0.154701

 




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