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++

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// 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;
}

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Java














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// 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 



















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Python3














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# 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 



















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C#














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// 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 



















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PHP














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<?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 


?> 



















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Output:


The ratio is 2 : 3






          
          
          
          

            

    

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