Check whether a given point lies inside a triangle or not | GeeksforGeeks

Check whether a given point lies inside a triangle or not | GeeksforGeeks
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.

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.

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);

}

Math proof
Area of the Triangle Formed by Three co-ordinate Points | Area of the Triangle
http://www.mathopenref.com/coordtrianglearea.html

Since the area of a trapezium = 1/2 × the sum of the parallel sides × the perpendicular distance between them,
Hence, the area of the triangle ABC = ∆ABC
= area of the trapezium ALNC + area of the trapezium CNMB - area of the trapezium ALMB
= 1/2 ∙ (AL + NC) . LN + 1/2 ∙ (CN + BM) ∙ NM - 1/2 ∙ (AL + BM).LM
= 1/2 ∙ (y₁ + y₃) (x₃ - x₁) + 1/2 ∙ (y₃ + y₂) (x₂ - x₃) - 1/2 ∙ (y₁ + y₂) (x₂ – x₁) // easier to understand
= 1/2 ∙ [x₁ y₂ – y₁ x₂ + x₂ y₃ - y₂ x₃ + x₃ y₁ – y₃ x₁]
= 1/2 [x₁ (y₂ - y₃) + x₂ (y₃ – y₁) + x₃ (y₁ – y₂)] sq. units. 

http://www.crazyforcode.com/check-point-lies-triangle/
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