Razón de la distancia entre los centros de las circunferencias y el punto de intersección de dos tangentes comunes directas a las circunferencias

Dadas dos circunferencias de radios dados, tales que las circunferencias no se tocan, la tarea es encontrar la razón de la distancia entre los centros de las circunferencias y el punto de intersección de dos tangentes comunes directas a las circunferencias.

Ejemplos: 

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

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

Enfoque : 

• Sean los radios de los círculos r ₁ y r ₂ y los centros C ₁ y C ₂ respectivamente.
• Sea P el punto de intersección de dos tangentes comunes directas a las circunferencias, y A ₁ & A ₂ el punto de contacto de las tangentes con las circunferencias.
• En el triángulo PC ₁ A ₁ y el triángulo PC ₂ A ₂ ,  el
  ángulo C ₁ A ₁ P = ángulo C ₂ A ₂ P = 90 grados { la línea que une el centro del círculo con el punto de contacto forma un ángulo de 90 grados con el tangente }, 
  también, ángulo A ₁ PC ₁ = ángulo A ₂ PC ₂ 
  entonces, ángulo A ₁ C ₁ P = ángulo A ₂ C ₂P 
  como ángulos son iguales, los triángulos PC ₁ A ₁ y PC ₂ A ₂ son similares .
• Entonces, debido a la similitud de los triángulos, 
  C ₁ P/C ₂ P = C ₁ A ₁ /C ₂ A ₂ = r ₁ /r ₂

La razón de la distancia entre los centros de los círculos y el punto de intersección de dos tangentes comunes directas a los círculos = radio del primer círculo: radio del segundo círculo

A continuación se muestra la implementación del enfoque anterior: 

// 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 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

# 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# 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 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
?>

<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>

The ratio is 2 : 3

Complejidad del tiempo: O(log(min(a, b)))

Espacio auxiliar: O(log(min(a, b)))