# 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.

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

The program needs to read the values of three coordinates

A(x1,y1)
B(x2,y2)
C(x3,y3)
as well as another coordinate P(x,y) and determine whether this point is inside a triangle formed from the 3 point above.

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.

Implementation

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  */
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);
}

Time : O(1)

Space: O(1)

### One Thought on “Check Whether a Given Point Lies inside a Triangle or not”

1. shruthi on July 29, 2016 at 11:01 am said:

why is area taken here?
and also how to determine in wich triangle of(a1,a2,a3) x and y points lye.