Ratio of the distance between the centers of the circles and the point of intersection of two direct common tangents to the circles

Given two circles of given radii, such that the circles don’t touch each other, the task is 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.

Examples:

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

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

Approach:

Let the radii of the circles be r₁ & r₂ and the centres be C₁ & C₂ respectively.
Let P be the point of intersection of two direct common tangents to the circles, and A₁ & A₂ be the point of contact of the tangents with the circles.
In triangle PC₁A₁ & triangle PC₂A₂,
angle C₁A₁P = angle C₂A₂P = 90 deg { line joining the center of the circle to the point of contact makes an angle of 90 degree with the tangent },
also, angle A₁PC₁ = angle A₂PC₂
so, angle A₁C₁P = angle A₂C₂P
as angles are same, triangles PC₁A₁ & PC₂A₂ are similiar.
So, due to similiarity of the triangles,
C₁P/C₂P = C₁A₁/C₂A₂ = r₁/r₂

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

Below is the implementation of the above approach:

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 <bits/stdc++.h> 
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 

PHP

<?php 
// PHP 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  
function GCD($a, $b)  
{  
    return ($b != 0 ? GCD($b, $a % $b) : $a);  
}  
  
// Function to find the ratio  
function ratiotang($r1, $r2)  
{  
    echo "The ratio is ", $r1 / GCD($r1, $r2), 
                " : ", $r2 / GCD($r1, $r2) ; 
}  
  
// Driver code  
$r1 = 4; $r2 = 6;  
ratiotang($r1, $r2);  
  
// This code is contributed by AnkitRai01 
?> 

Output:

The ratio is 2 : 3

My Personal Notes

C++

Java

Python3

C#

PHP

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