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

Check whether a given point lies inside a triangle or not - Geometric Algorithm - Given three corner points of a triangle, and one more point P.

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.

``````#include <stdio.h>
#include <stdlib.h>

/* 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;
}``````

Ouptut:

` Inside`

#### About the author #### Venkatesan Prabu

Wikitechy Founder, Author, International Speaker, and Job Consultant. My role as the CEO of Wikitechy, I help businesses build their next generation digital platforms and help with their product innovation and growth strategy. I'm a frequent speaker at tech conferences and events.