Given three non-collinear integral points in XY plane, find the number of integral points inside the triangle formed by the three points. (A point in XY plane is said to be integral/lattice point if both its co-ordinates are integral).

**Example:**

Input: p = (0, 0), q = (0, 5) and r = (5,0) Output: 6 Explanation: The points (1,1) (1,2) (1,3) (2,1) (2,2) and (3,1) are the integral points inside the triangle.

We can use the Pick’s theorem, which states that the following equation holds true for a simple Polygon.

Pick's Theeorem: A = I + (B/2) -1A==> Area of PolygonB==> Number of integral points on edges of polygonI==> Number of integral points inside the polygon Using the above formula, we can deduce, I = (2A - B + 2) / 2

We can find **A** (area of triangle) using below Shoelace formula.

A = 1/2 * abs(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))

**How to find B (number of integral points on edges of a triangle)?**

We can find the number of integral points between any two vertex (V1, V2) of the triangle using the following algorithm.

1. If the edge formed by joining V1 and V2 is parallel to the X-axis, then the number of integral points between the vertices is : abs(V1.x - V2.x) - 1 2. Similarly, if edge is parallel to the Y-axis, then the number of integral points in between is : abs(V1.y - V2.y) - 1 3. Else, we can find the integral points between the vertices using below formula:GCD(abs(V1.x-V2.x), abs(V1.y-V2.y)) - 1The above formula is a well known fact and can be verified using simple geometry. (Hint: Shift the edge such that one of the vertex lies at the Origin.) Please refer below link for detailed explanation. https://www.geeksforgeeks.org/number-integral-points-two-points/

Below is the implementation of the above algorithm.

## C++

`// C++ program to find Integral points inside a triangle` `#include<bits/stdc++.h>` `using` `namespace` `std;` `// Class to represent an Integral point on XY plane.` `class` `Point` `{` `public` `:` ` ` `int` `x, y;` ` ` `Point(` `int` `a=0, ` `int` `b=0):x(a),y(b) {}` `};` `//utility function to find GCD of two numbers` `// GCD of a and b` `int` `gcd(` `int` `a, ` `int` `b)` `{` ` ` `if` `(b == 0)` ` ` `return` `a;` ` ` `return` `gcd(b, a%b);` `}` `// Finds the no. of Integral points between` `// two given points.` `int` `getBoundaryCount(Point p,Point q)` `{` ` ` `// Check if line parallel to axes` ` ` `if` `(p.x==q.x)` ` ` `return` `abs` `(p.y - q.y) - 1;` ` ` `if` `(p.y == q.y)` ` ` `return` `abs` `(p.x - q.x) - 1;` ` ` `return` `gcd(` `abs` `(p.x-q.x), ` `abs` `(p.y-q.y)) - 1;` `}` `// Returns count of points inside the triangle` `int` `getInternalCount(Point p, Point q, Point r)` `{` ` ` `// 3 extra integer points for the vertices` ` ` `int` `BoundaryPoints = getBoundaryCount(p, q) +` ` ` `getBoundaryCount(p, r) +` ` ` `getBoundaryCount(q, r) + 3;` ` ` `// Calculate 2*A for the triangle` ` ` `int` `doubleArea = ` `abs` `(p.x*(q.y - r.y) + q.x*(r.y - p.y) +` ` ` `r.x*(p.y - q.y));` ` ` `// Use Pick's theorem to calculate the no. of Interior points` ` ` `return` `(doubleArea - BoundaryPoints + 2)/2;` `}` `// driver function to check the program.` `int` `main()` `{` ` ` `Point p(0, 0);` ` ` `Point q(5, 0);` ` ` `Point r(0, 5);` ` ` `cout << ` `"Number of integral points inside given triangle is "` ` ` `<< getInternalCount(p, q, r);` ` ` `return` `0;` `}` |

## Java

`// Java program to find Integral points inside a triangle` `// Class to represent an Integral point on XY plane.` `class` `Point` `{` ` ` `int` `x, y;` ` ` `public` `Point(` `int` `a, ` `int` `b)` ` ` `{` ` ` `x = a;` ` ` `y = b;` ` ` `}` `}` `class` `GFG` `{` ` ` `// utility function to find GCD of two numbers` ` ` `// GCD of a and b` ` ` `static` `int` `gcd(` `int` `a, ` `int` `b)` ` ` `{` ` ` `if` `(b == ` `0` `)` ` ` `return` `a;` ` ` `return` `gcd(b, a % b);` ` ` `}` ` ` `// Finds the no. of Integral points between` ` ` `// two given points.` ` ` `static` `int` `getBoundaryCount(Point p, Point q)` ` ` `{` ` ` `// Check if line parallel to axes` ` ` `if` `(p.x == q.x)` ` ` `return` `Math.abs(p.y - q.y) - ` `1` `;` ` ` `if` `(p.y == q.y)` ` ` `return` `Math.abs(p.x - q.x) - ` `1` `;` ` ` `return` `gcd(Math.abs(p.x - q.x),` ` ` `Math.abs(p.y - q.y)) - ` `1` `;` ` ` `}` ` ` `// Returns count of points inside the triangle` ` ` `static` `int` `getInternalCount(Point p, Point q, Point r)` ` ` `{` ` ` `// 3 extra integer points for the vertices` ` ` `int` `BoundaryPoints = getBoundaryCount(p, q) +` ` ` `getBoundaryCount(p, r) +` ` ` `getBoundaryCount(q, r) + ` `3` `;` ` ` `// Calculate 2*A for the triangle` ` ` `int` `doubleArea = Math.abs(p.x * (q.y - r.y) +` ` ` `q.x * (r.y - p.y) +` ` ` `r.x * (p.y - q.y));` ` ` `// Use Pick's theorem to calculate` ` ` `// the no. of Interior points` ` ` `return` `(doubleArea - BoundaryPoints + ` `2` `) / ` `2` `;` ` ` `}` ` ` `// Driver Code` ` ` `public` `static` `void` `main(String[] args)` ` ` `{` ` ` `Point p = ` `new` `Point(` `0` `, ` `0` `);` ` ` `Point q = ` `new` `Point(` `5` `, ` `0` `);` ` ` `Point r = ` `new` `Point(` `0` `, ` `5` `);` ` ` `System.out.println(` `"Number of integral points"` `+` ` ` `" inside given triangle is "` `+` ` ` `getInternalCount(p, q, r));` ` ` `}` `}` `// This code is contributed by Vivek Kumar Singh` |

## Python3

`# Python3 program to find Integral` `# points inside a triangle` `# Class to represent an Integral` `# point on XY plane.` `class` `Point:` ` ` `def` `__init__(` `self` `, x, y):` ` ` `self` `.x ` `=` `x` ` ` `self` `.y ` `=` `y` ` ` `# Utility function to find GCD of` `# two numbers GCD of a and b` `def` `gcd(a, b):` ` ` `if` `(b ` `=` `=` `0` `):` ` ` `return` `a` ` ` ` ` `return` `gcd(b, a ` `%` `b)` `# Finds the no. of Integral points` `# between two given points` `def` `getBoundaryCount(p, q):` ` ` ` ` `# Check if line parallel to axes` ` ` `if` `(p.x ` `=` `=` `q.x):` ` ` `return` `abs` `(p.y ` `-` `q.y) ` `-` `1` ` ` `if` `(p.y ` `=` `=` `q.y):` ` ` `return` `abs` `(p.x ` `-` `q.x) ` `-` `1` ` ` `return` `gcd(` `abs` `(p.x ` `-` `q.x),` ` ` `abs` `(p.y ` `-` `q.y)) ` `-` `1` `# Returns count of points inside the triangle` `def` `getInternalCount(p, q, r):` ` ` `# 3 extra integer points for the vertices` ` ` `BoundaryPoints ` `=` `(getBoundaryCount(p, q) ` `+` ` ` `getBoundaryCount(p, r) ` `+` ` ` `getBoundaryCount(q, r) ` `+` `3` `)` ` ` `# Calculate 2*A for the triangle` ` ` `doubleArea ` `=` `abs` `(p.x ` `*` `(q.y ` `-` `r.y) ` `+` ` ` `q.x ` `*` `(r.y ` `-` `p.y) ` `+` ` ` `r.x ` `*` `(p.y ` `-` `q.y))` ` ` `# Use Pick's theorem to calculate` ` ` `# the no. of Interior points` ` ` `return` `(doubleArea ` `-` `BoundaryPoints ` `+` `2` `) ` `/` `/` `2` `# Driver code` `if` `__name__` `=` `=` `"__main__"` `:` ` ` ` ` `p ` `=` `Point(` `0` `, ` `0` `)` ` ` `q ` `=` `Point(` `5` `, ` `0` `)` ` ` `r ` `=` `Point(` `0` `, ` `5` `)` ` ` ` ` `print` `(` `"Number of integral points "` ` ` `"inside given triangle is "` `,` ` ` `getInternalCount(p, q, r))` ` ` `# This code is contributed by rutvik_56` |

## C#

`// C# program to find Integral points` `// inside a triangle` `using` `System;` `// Class to represent an Integral point` `// on XY plane.` `public` `class` `Point` `{` ` ` `public` `int` `x, y;` ` ` `public` `Point(` `int` `a, ` `int` `b)` ` ` `{` ` ` `x = a;` ` ` `y = b;` ` ` `}` `}` `class` `GFG` `{` ` ` `// utility function to find GCD of` ` ` `// two numbers a and b` ` ` `static` `int` `gcd(` `int` `a, ` `int` `b)` ` ` `{` ` ` `if` `(b == 0)` ` ` `return` `a;` ` ` `return` `gcd(b, a % b);` ` ` `}` ` ` `// Finds the no. of Integral points between` ` ` `// two given points.` ` ` `static` `int` `getBoundaryCount(Point p, Point q)` ` ` `{` ` ` `// Check if line parallel to axes` ` ` `if` `(p.x == q.x)` ` ` `return` `Math.Abs(p.y - q.y) - 1;` ` ` `if` `(p.y == q.y)` ` ` `return` `Math.Abs(p.x - q.x) - 1;` ` ` `return` `gcd(Math.Abs(p.x - q.x),` ` ` `Math.Abs(p.y - q.y)) - 1;` ` ` `}` ` ` `// Returns count of points inside the triangle` ` ` `static` `int` `getInternalCount(Point p, Point q, Point r)` ` ` `{` ` ` `// 3 extra integer points for the vertices` ` ` `int` `BoundaryPoints = getBoundaryCount(p, q) +` ` ` `getBoundaryCount(p, r) +` ` ` `getBoundaryCount(q, r) + 3;` ` ` `// Calculate 2*A for the triangle` ` ` `int` `doubleArea = Math.Abs(p.x * (q.y - r.y) +` ` ` `q.x * (r.y - p.y) +` ` ` `r.x * (p.y - q.y));` ` ` `// Use Pick's theorem to calculate` ` ` `// the no. of Interior points` ` ` `return` `(doubleArea - BoundaryPoints + 2) / 2;` ` ` `}` ` ` `// Driver Code` ` ` `public` `static` `void` `Main(String[] args)` ` ` `{` ` ` `Point p = ` `new` `Point(0, 0);` ` ` `Point q = ` `new` `Point(5, 0);` ` ` `Point r = ` `new` `Point(0, 5);` ` ` `Console.WriteLine(` `"Number of integral points"` `+` ` ` `" inside given triangle is "` `+` ` ` `getInternalCount(p, q, r));` ` ` `}` `}` `// This code is contributed by 29AjayKumar` |

**Output:**

Number of integral points inside given triangle is 6

This article is contributed by **Ashutosh Kumar**. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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