To verify `Pappus Theorem`.

pappusth.py

"""In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of 
collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of 
line pairs Ab and aB, Ac and aC, Bc and bC are collinear, lying on the Pappus line. 
"""

from sympy import *
from sympy.geometry import *
l1 = Line(Point(0, 0), Point(5, 6))
l2 = Line(Point(0, 0), Point(2, -2))

def subs_point(l, val):
  """Take an arbitrary point and make it a fixed point."""
  t = Symbol('t', real=True)
  ap = l.arbitrary_point()
  return Point(ap.x.subs(t, val), ap.y.subs(t, val))


p11 = subs_point(l1, 5)
p12 = subs_point(l1, 6)
p13 = subs_point(l1, 11)

p21 = subs_point(l2, -1)
p22 = subs_point(l2, 2)
p23 = subs_point(l2, 13)

ll1 = Line(p11, p22)
ll2 = Line(p11, p23)
ll3 = Line(p12, p21)
ll4 = Line(p12, p23)
ll5 = Line(p13, p21)
ll6 = Line(p13, p22)


pp1 = intersection(ll1, ll3)[0]
pp2 = intersection(ll2, ll5)[0]
pp3 = intersection(ll4, ll6)[0]

print (Point.is_collinear(pp1, pp2, pp3)) # Prints true