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 centers 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 similar.
• So, due to similarity of the triangles, 
  C₁P/C₂P = C₁A₁/C₂A₂ = r₁/r₂

The ratio of the distance between the centers 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
?>

Javascript

<script>
// javascript 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)
{
    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
</script>

Output:

The ratio is 2 : 3

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

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