圆心之间的距离与圆的两条直接公切线的交点的比值

给定两个给定半径的圆，使圆彼此不接触，任务是找到圆心之间的距离与圆的两个直接公切线的交点的比值。
例子：

Input: r1 = 4, r2 = 6 
Output: 2:3

Input: r1 = 22, r2 = 43
Output: 22:43

方法：

• 设圆的半径分别为r ₁ & r ₂ ，圆心分别为C ₁ & C ₂ 。
• 设P为圆的两条直接公切线的交点， A ₁ & A ₂为切线与圆的接触点。
• 在三角形PC ₁ A ₁和三角形PC ₂ A _{2 中}，
  角C ₁ A ₁ P = 角C ₂ A ₂ P = 90 度{ 连接圆心和接触点的线与切线成 90 度角 },
  同样，角度A ₁ PC ₁ = 角度A ₂ PC ₂
  所以，角A ₁ C ₁ P = 角A ₂ C ₂ P
  由于角度相同，三角形PC ₁ A ₁和PC ₂ A ₂是相似的。
• 所以，由于三角形的相似性，
  C ₁ P/C ₂ P = C ₁ A ₁ /C ₂ A ₂ = r ₁ /r ₂

The ratio of the distance between the centres of the circles and the point of intersection of two direct common tangents to the circles = radius of the first circle : radius of the second circle

下面是上述方法的实现：

C++

// C++ program to find the ratio of the distance
// between the centers of the circles
// and the point of intersection of
// two direct common tangents to the circles
// which do not touch each other
 
#include 
using namespace std;
 
// Function to find the GCD
int GCD(int a, int b)
{
    return (b != 0 ? GCD(b, a % b) : a);
}
 
// Function to find the ratio
void ratiotang(int r1, int r2)
{
    cout << "The ratio is "
         << r1 / GCD(r1, r2)
         << " : "
         << r2 / GCD(r1, r2)
         << endl;
}
 
// Driver code
int main()
{
    int r1 = 4, r2 = 6;
    ratiotang(r1, r2);
    return 0;
}

Java

// Java program to find the ratio of the distance
// between the centers of the circles
// and the point of intersection of
// two direct common tangents to the circles
// which do not touch each other
class GFG {
 
    // Function to find the GCD
    static int GCD(int a, int b)
    {
        return (b != 0 ? GCD(b, a % b) : a);
    }
 
    // Function to find the ratio
    static void ratiotang(int r1, int r2)
    {
        System.out.println("The ratio is "
                           + r1 / GCD(r1, r2)
                           + " : "
                           + r2 / GCD(r1, r2));
    }
 
    // Driver code
    public static void main(String args[])
    {
        int r1 = 4, r2 = 6;
        ratiotang(r1, r2);
    }
}
 
// This code has been contributed by 29AjayKumar

Python3

# Python 3 program to find the ratio
# of the distance between the centers
# of the circles and the point of intersection
# of two direct common tangents to the circles
# which do not touch each other
 
# Function to find the GCD
from math import gcd
 
# Function to find the ratio
def ratiotang(r1, r2):
    print("The ratio is", int(r1 / gcd(r1, r2)), ":",
                          int(r2 / gcd(r1, r2)))
 
# Driver code
if __name__ == '__main__':
    r1 = 4
    r2 = 6
    ratiotang(r1, r2)
 
# This code is contributed by
# Surendra_Gangwar

C#

// C# program to find the ratio of the distance
// between the centers of the circles
// and the point of intersection of
// two direct common tangents to the circles
// which do not touch each other
using System;
     
class GFG
{
 
    // Function to find the GCD
    static int GCD(int a, int b)
    {
        return (b != 0 ? GCD(b, a % b) : a);
    }
 
    // Function to find the ratio
    static void ratiotang(int r1, int r2)
    {
        Console.WriteLine("The ratio is "
                        + r1 / GCD(r1, r2)
                        + " : "
                        + r2 / GCD(r1, r2));
    }
 
    // Driver code
    public static void Main(String[] args)
    {
        int r1 = 4, r2 = 6;
        ratiotang(r1, r2);
    }
}
 
// This code contributed by Rajput-Ji

Javascript

// javascript program to find the ratio of the distance
// between the centers of the circles
// and the povar of intersection of
// two direct common tangents to the circles
// which do not touch each other
 
// Function to find the GCD
function GCD(a , b)
{
    return (b != 0 ? GCD(b, a % b) : a);
}
 
// Function to find the ratio
function ratiotang(r1 , r2)
{
    document.write("The ratio is "
                       + r1 / GCD(r1, r2)
                       + " : "
                       + r2 / GCD(r1, r2));
}
 
// Driver code
var r1 = 4, r2 = 6;
ratiotang(r1, r2);
 
// This code is contributed by Princi Singh

输出：

The ratio is 2 : 3