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 = 8Output :1:2Input :r1 = 5, r2 = 13Output :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 center 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 similiar. - So, due to similiarity 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; ` `} ` |

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## 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 ` |

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## 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 ` |

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## 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. ` |

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## 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 ` `?> ` |

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**Output:**

The ratio is 1:2

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