Ratio of the distance between the centers of the circles and the point of intersection of two transverse 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 centres of the circles and the point of intersection of two transverse common tangents to the circles.
Examples:

Input :r1 = 4, r2 = 8 
Output :1:2

Input :r1 = 5, r2 = 13
Output :5:13

Approach: 

• Let the radii of the circles be r1 & r2 and C1 & C2 respectively.
• Let P be the point of intersection of two transverse 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 centre of the circle to the point of contact makes an angle of 90 degree with the tangent }, 
  also, angle A1PC1 = angle A2PC2{vertically opposite angles are always equal} 
  so, angle A1C1P = angle A2C2P
  as angles are same, triangles PC1A1 & PC2A2 are similar.
• So, due to similarity of the triangles, 
  C1P/C2P = C1A1/C2A2 = r1/r2

C++

// C++ program to find the ratio
// of the distance between the centres of the circles
// and the point of intersection
// of two transverse common tangents
// to the circles which do not touch each other

#include <bits/stdc++.h>
using namespace std;

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 = 8;
    ratiotang(r1, r2);
    return 0;
}

Java

// Java program to find the ratio
// of the distance between the centres of the circles
// and the point of intersection
// of two transverse common tangents
// to the circles which do not touch each other

import java.io.*;

class GFG{

    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 = 8;
        ratiotang(r1, r2);
    }
}

// This code is contributed by NamrataSrivastava1

Python

# Python3 program to find the ratio
# of the distance between the centres of the circles
# and the point of intersection
# of two transverse common tangents
# to the circles which do not touch each other

def GCD(a, b):
    if(b!=0):
        return GCD(b, a%b);
    else:
        return a;

# Function to find the ratio
def ratiotang(r1, r2):

    print("The ratio is", r1 // GCD(r1, r2),
                     ":", r2 // GCD(r1, r2));

# Driver code
r1 = 4; r2 = 8;
ratiotang(r1, r2);

# This code is contributed by Code_Mech

C#

// C# program to find the ratio
// of the distance between the centres of the circles
// and the point of intersection
// of two transverse common tangents
// to the circles which do not touch each other
using System;

class GFG
{

    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
    static public void Main ()
    {

        int r1 = 4, r2 = 8;
        ratiotang(r1, r2);
    }
}

// This code is contributed by Tushil.

PHP

<?php
// PHP program to find the ratio
// of the distance between the centres of the circles
// and the point of intersection
// of two transverse common tangents
// to the circles which do not touch each other

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;
$r2 = 8;
ratiotang($r1, $r2);

// This code is contributed by AnkitRai01
?>

Javascript

<script>
// javascript program to find the ratio
// of the distance between the centres of the circles
// and the point of intersection
// of two transverse common tangents
// to the circles which do not touch each other
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 = 8;
ratiotang(r1, r2);

// This code is contributed by Princi Singh
</script>

Output:

The ratio is 1:2

Time Complexity: O(log(min(a, b)))

Auxiliary Space: O(log(min(a, b)))