笛卡尔圆定理与实现

在几何学中，笛卡尔定理指出，对于每四个相互相切的圆，圆的半径满足某个二次方程。可以构造与三个给定的相互相切的圆相切的第四个圆。
笛卡尔圆定理帮助我们在有4 个正整数半径为r1 、 r2 、 r3和r4的圆时找到圆的半径，如下图所示。它找到由半径为r1 、 r2 、 r3的三个圆形成的圆的半径r4 ，如下图所示。
（请注意，下图中的圆圈彼此相切。）

例子：

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

该定理说，这些圆的半径或“曲率”的倒数

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

一个圆可以被其余三个圆包围，如图所示，在这种情况下，相应的曲率$(k_4 = -1/r_4)$这里被认为是否定的，并且上述关系仍然成立。
如果$k_1, k_2, k_3$已知，可以求解 k4，
$k_4 = k_1 + k_2 + k_3 \pm 2\sqrt{k_1 k_2 + k_2 k_3 + k_1 k_3}$
在求解上述方程时，我们得到第四个圆的半径。求第四个圆的半径的公式是：

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

因此，如果 r1、r2 和 r3 已知，则可以使用上述公式轻松计算 r4。
下面是上述方法的实现：

CPP

// C++ implementation of the
// above formulae
#include 
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

// 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.

输出：

The radius of fourth circle: 0.154701

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