Find if a point is within a triangle

Task

Assume points are on a plane defined by (x, y) ewal number coordinates.

Given a point P(x, y) and a triangle formed by points A, B, and C, determine if P is within triangle ABC.

You may use any algorithm.

Bonus: explain why the algorithm you chose works.

Related tasks

Determine_if_two_triangles_overlap

Also see

Discussion of several methods. [[1]]
Determine if a point is in a polygon [[2]]
Triangle based coordinate systems [[3]]
Wolfram entry [[4]]

Python

<lang python> """ find if point is in a triangle """

from sympy.geometry import Point, Triangle

def sign(pt1, pt2, pt3):

   """ which side of plane cut by line (pt2, pt3) is pt1 on? """
   return (pt1.x - pt3.x) * (pt2.y - pt3.y) - (pt2.x - pt3.x) * (pt1.y - pt3.y)

def iswithin(point, pt1, pt2, pt3):

   """ 
   Determine if point is within triangle formed by points p1, p2, p3.
   If so, the point will be on the same side of each of the half planes
   defined by vectors p1p2, p2p3, and p3p1. zval is positive if outside,
   negative if inside such a plane. All should be positive or all negative
   if point is within the triangle.
   """
   zval1 = sign(point, pt1, pt2)
   zval2 = sign(point, pt2, pt3)
   zval3 = sign(point, pt3, pt1)
   notanyneg = zval1 >= 0 and zval2 >= 0 and zval3 >= 0
   notanypos = zval1 <= 0 and zval2 <= 0 and zval3 <= 0
   return notanyneg or notanypos

if __name__ == "__main__":

   POINTS = [Point(0, 0)]
   TRI = Triangle(Point(1.5, 2.4), Point(5.1, -3.1), Point(-3.8, 0.5))
   for pnt in POINTS:
       a, b, c = TRI.vertices
       isornot = "is" if iswithin(pnt, a, b, c) else "is not"
       print("Point ", pnt, isornot, "within the triangle ", TRI)

<lang>

Output:

Point  Point2D(0, 0) is within the triangle  Triangle(Point2D(3/2, 12/5), Point2D(51/10, -31/10), Point2D(-19/5, 1/2))