# 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 = 6Output:2:3Input:r1 = 22, r2 = 43Output:22:43

**Approach**:

- Let the radii of the circles be
**r**&_{1}**r**and the centres be_{2}**C**&_{1}**C**respectively._{2} - Let
**P**be the point of intersection of two direct common tangents to the circles, and**A**&_{1}**A**be the point of contact of the tangents with the circles._{2} - In triangle
**PC**& triangle_{1}A_{1}**PC**,_{2}A_{2}

angle**C**= angle_{1}A_{1}P**C**=_{2}A_{2}P**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**A**= angle_{1}PC_{1}**A**_{2}PC_{2}

so, angle**A**= angle_{1}C_{1}P**A**_{2}C_{2}P

as angles are same, triangles**PC**&_{1}A_{1}**PC**are_{2}A_{2}**similiar**. - So, due to similiarity of the triangles,

**C**_{1}P/C_{2}P = C_{1}A_{1}/C_{2}A_{2}= r_{1}/r_{2}

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; ` `} ` |

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## 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 ` |

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## 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 ` |

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## 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 ` |

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## 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 ` `?> ` |

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**Output:**

The ratio is 2 : 3

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- Check if two given circles touch or intersect each other
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