# Check whether a given point lies inside a triangle or not

Given three corner points of a triangle, and one more point P. Write a function to check whether P lies within the triangle or not.

For example, consider the following program, the function should return true for P(10, 15) and false for P'(30, 15)

```              B(10,30)
/
/
/
/   P         P'
/
A(0,0) ----------- C(20,0)
```

Solution:
Let the coordinates of three corners be (x1, y1), (x2, y2) and (x3, y3). And coordinates of the given point P be (x, y)

1) Calculate area of the given triangle, i.e., area of the triangle ABC in the above diagram. Area A = [ x1(y2 – y3) + x2(y3 – y1) + x3(y1-y2)]/2
2) Calculate area of the triangle PAB. We can use the same formula for this. Let this area be A1.
3) Calculate area of the triangle PBC. Let this area be A2.
4) Calculate area of the triangle PAC. Let this area be A3.
5) If P lies inside the triangle, then A1 + A2 + A3 must be equal to A.

## C++

 `#include ` `using` `namespace` `std; ` ` `  `/* A utility function to calculate area of triangle formed by (x1, y1),  ` `   ``(x2, y2) and (x3, y3) */`  `float` `area(``int` `x1, ``int` `y1, ``int` `x2, ``int` `y2, ``int` `x3, ``int` `y3) ` `{ ` `   ``return` `abs``((x1*(y2-y3) + x2*(y3-y1)+ x3*(y1-y2))/2.0); ` `} ` ` `  `/* A function to check whether point P(x, y) lies inside the triangle formed  ` `   ``by A(x1, y1), B(x2, y2) and C(x3, y3) */`  `bool` `isInside(``int` `x1, ``int` `y1, ``int` `x2, ``int` `y2, ``int` `x3, ``int` `y3, ``int` `x, ``int` `y) ` `{    ` `   ``/* Calculate area of triangle ABC */` `   ``float` `A = area (x1, y1, x2, y2, x3, y3); ` ` `  `   ``/* Calculate area of triangle PBC */`    `   ``float` `A1 = area (x, y, x2, y2, x3, y3); ` ` `  `   ``/* Calculate area of triangle PAC */`    `   ``float` `A2 = area (x1, y1, x, y, x3, y3); ` ` `  `   ``/* Calculate area of triangle PAB */`     `   ``float` `A3 = area (x1, y1, x2, y2, x, y); ` `   `  `   ``/* Check if sum of A1, A2 and A3 is same as A */`  `   ``return` `(A == A1 + A2 + A3); ` `} ` ` `  `/* Driver program to test above function */` `int` `main() ` `{ ` `   ``/* Let us check whether the point P(10, 15) lies inside the triangle  ` `      ``formed by A(0, 0), B(20, 0) and C(10, 30) */` `   ``if` `(isInside(0, 0, 20, 0, 10, 30, 10, 15)) ` `     ``printf` `(``"Inside"``); ` `   ``else` `     ``printf` `(``"Not Inside"``); ` ` `  `   ``return` `0; ` `} `

## Java

 `// JAVA Code for Check whether a given point  ` `// lies inside a triangle or not ` `import` `java.util.*; ` ` `  `class` `GFG { ` `     `  `    ``/* A utility function to calculate area of triangle  ` `       ``formed by (x1, y1) (x2, y2) and (x3, y3) */` `    ``static` `double` `area(``int` `x1, ``int` `y1, ``int` `x2, ``int` `y2, ` `                                        ``int` `x3, ``int` `y3) ` `    ``{ ` `       ``return` `Math.abs((x1*(y2-y3) + x2*(y3-y1)+ ` `                                    ``x3*(y1-y2))/``2.0``); ` `    ``} ` `      `  `    ``/* A function to check whether point P(x, y) lies ` `       ``inside the triangle formed by A(x1, y1), ` `       ``B(x2, y2) and C(x3, y3) */` `    ``static` `boolean` `isInside(``int` `x1, ``int` `y1, ``int` `x2, ` `                ``int` `y2, ``int` `x3, ``int` `y3, ``int` `x, ``int` `y) ` `    ``{    ` `       ``/* Calculate area of triangle ABC */` `        ``double` `A = area (x1, y1, x2, y2, x3, y3); ` `      `  `       ``/* Calculate area of triangle PBC */`   `        ``double` `A1 = area (x, y, x2, y2, x3, y3); ` `      `  `       ``/* Calculate area of triangle PAC */`   `        ``double` `A2 = area (x1, y1, x, y, x3, y3); ` `      `  `       ``/* Calculate area of triangle PAB */`    `        ``double` `A3 = area (x1, y1, x2, y2, x, y); ` `        `  `       ``/* Check if sum of A1, A2 and A3 is same as A */` `        ``return` `(A == A1 + A2 + A3); ` `    ``} ` `     `  `    ``/* Driver program to test above function */` `    ``public` `static` `void` `main(String[] args)  ` `    ``{ ` `        ``/* Let us check whether the point P(10, 15) ` `           ``lies inside the triangle formed by  ` `           ``A(0, 0), B(20, 0) and C(10, 30) */` `       ``if` `(isInside(``0``, ``0``, ``20``, ``0``, ``10``, ``30``, ``10``, ``15``)) ` `           ``System.out.println(``"Inside"``); ` `       ``else` `           ``System.out.println(``"Not Inside"``); ` `             `  `    ``} ` `} ` ` `  `// This code is contributed by Arnav Kr. Mandal. `

/div>

## Python

 `# A utility function to calculate area  ` `# of triangle formed by (x1, y1),  ` `# (x2, y2) and (x3, y3) ` ` `  `def` `area(x1, y1, x2, y2, x3, y3): ` ` `  `    ``return` `abs``((x1 ``*` `(y2 ``-` `y3) ``+` `x2 ``*` `(y3 ``-` `y1)  ` `                ``+` `x3 ``*` `(y1 ``-` `y2)) ``/` `2.0``) ` ` `  ` `  `# A function to check whether point P(x, y) ` `# lies inside the triangle formed by  ` `# A(x1, y1), B(x2, y2) and C(x3, y3)  ` `def` `isInside(x1, y1, x2, y2, x3, y3, x, y): ` ` `  `    ``# Calculate area of triangle ABC ` `    ``A ``=` `area (x1, y1, x2, y2, x3, y3) ` ` `  `    ``# Calculate area of triangle PBC  ` `    ``A1 ``=` `area (x, y, x2, y2, x3, y3) ` `     `  `    ``# Calculate area of triangle PAC  ` `    ``A2 ``=` `area (x1, y1, x, y, x3, y3) ` `     `  `    ``# Calculate area of triangle PAB  ` `    ``A3 ``=` `area (x1, y1, x2, y2, x, y) ` `     `  `    ``# Check if sum of A1, A2 and A3  ` `    ``# is same as A ` `    ``if``(A ``=``=` `A1 ``+` `A2 ``+` `A3): ` `        ``return` `True` `    ``else``: ` `        ``return` `False` ` `  `# Driver program to test above function ` `# Let us check whether the point P(10, 15) ` `# lies inside the triangle formed by  ` `# A(0, 0), B(20, 0) and C(10, 30) ` `if` `(isInside(``0``, ``0``, ``20``, ``0``, ``10``, ``30``, ``10``, ``15``)): ` `    ``print``(``'Inside'``) ` `else``: ` `    ``print``(``'Not Inside'``)  ` ` `  `# This code is contributed by Danish Raza `

## C#

 `// C# Code to Check whether a given point ` `// lies inside a triangle or not ` `using` `System; ` ` `  `class` `GFG { ` ` `  `    ``/* A utility function to calculate area of triangle  ` `    ``formed by (x1, y1) (x2, y2) and (x3, y3) */` `    ``static` `double` `area(``int` `x1, ``int` `y1, ``int` `x2,  ` `                       ``int` `y2, ``int` `x3, ``int` `y3) ` `    ``{ ` `        ``return` `Math.Abs((x1 * (y2 - y3) + ` `                         ``x2 * (y3 - y1) +  ` `                         ``x3 * (y1 - y2)) / 2.0); ` `    ``} ` ` `  `    ``/* A function to check whether point P(x, y) lies ` `    ``inside the triangle formed by A(x1, y1), ` `    ``B(x2, y2) and C(x3, y3) */` `    ``static` `bool` `isInside(``int` `x1, ``int` `y1, ``int` `x2, ` `                         ``int` `y2, ``int` `x3, ``int` `y3,  ` `                         ``int` `x, ``int` `y) ` `    ``{ ` `        ``/* Calculate area of triangle ABC */` `        ``double` `A = area(x1, y1, x2, y2, x3, y3); ` ` `  `        ``/* Calculate area of triangle PBC */` `        ``double` `A1 = area(x, y, x2, y2, x3, y3); ` ` `  `        ``/* Calculate area of triangle PAC */` `        ``double` `A2 = area(x1, y1, x, y, x3, y3); ` ` `  `        ``/* Calculate area of triangle PAB */` `        ``double` `A3 = area(x1, y1, x2, y2, x, y); ` ` `  `        ``/* Check if sum of A1, A2 and A3 is same as A */` `        ``return` `(A == A1 + A2 + A3); ` `    ``} ` ` `  `/* Driver program to test above function */` `public` `static` `void` `Main() ` `{ ` `    ``/* Let us check whether the point P(10, 15) ` `    ``lies inside the triangle formed by  ` `    ``A(0, 0), B(20, 0) and C(10, 30) */` `    ``if` `(isInside(0, 0, 20, 0, 10, 30, 10, 15)) ` `        ``Console.WriteLine(``"Inside"``); ` `    ``else` `        ``Console.WriteLine(``"Not Inside"``); ` `} ` `} ` ` `  `// This code is contributed by vt_m. `

## PHP

 ` `

Ouptut:

` Inside `

Exercise: Given coordinates of four corners of a rectangle, and a point P. Write a function to check whether P lies inside the given rectangle or not.