code: def test_point():
      x = Symbol("x", real=True)
      y = Symbol("y", real=True)
      x1 = Symbol("x1", real=True)
      x2 = Symbol("x2", real=True)
      y1 = Symbol("y1", real=True)
      y2 = Symbol("y2", real=True)
      half = S.Half
      p1 = Point(x1, x2)
      p2 = Point(y1, y2)
      p3 = Point(0, 0)
      p4 = Point(1, 1)
      p5 = Point(0, 1)
      line = Line(Point(1, 0), slope=1)
  
      assert p1 in p1
      assert p1 not in p2
      assert p2.y == y2
      assert (p3 + p4) == p4
      assert (p2 - p1) == Point(y1 - x1, y2 - x2)
      assert -p2 == Point(-y1, -y2)
      raises(TypeError, lambda: Point(1))
      raises(ValueError, lambda: Point([1]))
      raises(ValueError, lambda: Point(3, I))
      raises(ValueError, lambda: Point(2 * I, I))
      raises(ValueError, lambda: Point(3 + I, I))
  
      assert Point(34.05, sqrt(3)) == Point(Rational(681, 20), sqrt(3))
      assert Point.midpoint(p3, p4) == Point(half, half)
      assert Point.midpoint(p1, p4) == Point(
          half + half * x1, half + half * x2
      )
      assert Point.midpoint(p2, p2) == p2
      assert p2.midpoint(p2) == p2
      assert p1.origin == Point(0, 0)
  
      assert Point.distance(p3, p4) == sqrt(2)
      assert Point.distance(p1, p1) == 0
      assert Point.distance(p3, p2) == sqrt(p2.x**2 + p2.y**2)
      raises(TypeError, lambda: Point.distance(p1, 0))
      raises(TypeError, lambda: Point.distance(p1, GeometryEntity()))
  
      # distance should be symmetric
      assert p1.distance(line) == line.distance(p1)
      assert p4.distance(line) == line.distance(p4)
  
      assert Point.taxicab_distance(p4, p3) == 2
  
      assert Point.canberra_distance(p4, p5) == 1
      raises(ValueError, lambda: Point.canberra_distance(p3, p3))
  
      p1_1 = Point(x1, x1)
      p1_2 = Point(y2, y2)
      p1_3 = Point(x1 + 1, x1)
      assert Point.is_collinear(p3)
  
      with warns(UserWarning, test_stacklevel=False):
          assert Point.is_collinear(p3, Point(p3, dim=4))
      assert p3.is_collinear()
      assert Point.is_collinear(p3, p4)
      assert Point.is_collinear(p3, p4, p1_1, p1_2)
      assert Point.is_collinear(p3, p4, p1_1, p1_3) is False
      assert Point.is_collinear(p3, p3, p4, p5) is False
  
      raises(TypeError, lambda: Point.is_collinear(line))
      raises(TypeError, lambda: p1_1.is_collinear(line))
  
      assert p3.intersection(Point(0, 0)) == [p3]
      assert p3.intersection(p4) == []
      assert p3.intersection(line) == []
      with warns(UserWarning, test_stacklevel=False):
          assert Point.intersection(Point(0, 0, 0), Point(0, 0)) == [
              Point(0, 0, 0)
          ]
  
      x_pos = Symbol("x", positive=True)
      p2_1 = Point(x_pos, 0)
      p2_2 = Point(0, x_pos)
      p2_3 = Point(-x_pos, 0)
      p2_4 = Point(0, -x_pos)
      p2_5 = Point(x_pos, 5)
      assert Point.is_concyclic(p2_1)
      assert Point.is_concyclic(p2_1, p2_2)
      assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_4)
      for pts in permutations((p2_1, p2_2, p2_3, p2_5)):
          assert Point.is_concyclic(*pts) is False
      assert Point.is_concyclic(p4, p4 * 2, p4 * 3) is False
      assert Point(0, 0).is_concyclic((1, 1), (2, 2), (2, 1)) is False
      assert (
          Point.is_concyclic(
              Point(0, 0, 0, 0),
              Point(1, 0, 0, 0),
              Point(1, 1, 0, 0),
              Point(1, 1, 1, 0),
          )
          is False
      )
  
      assert p1.is_scalar_multiple(p1)
      assert p1.is_scalar_multiple(2 * p1)
      assert not p1.is_scalar_multiple(p2)
      assert Point.is_scalar_multiple(Point(1, 1), (-1, -1))
      assert Point.is_scalar_multiple(Point(0, 0), (0, -1))
      # test when is_scalar_multiple can't be determined
      raises(
          Undecidable,
          lambda: Point.is_scalar_multiple(
              Point(sympify("x1%y1"), sympify("x2%y2")), Point(0, 1)
          ),
      )
  
      assert Point(0, 1).orthogonal_direction == Point(1, 0)
      assert Point(1, 0).orthogonal_direction == Point(0, 1)
  
      assert p1.is_zero is None
      assert p3.is_zero
      assert p4.is_zero is False
      assert p1.is_nonzero is None
      assert p3.is_nonzero is False
      assert p4.is_nonzero
  
      assert p4.scale(2, 3) == Point(2, 3)
      assert p3.scale(2, 3) == p3
  
      assert p4.rotate(pi, Point(0.5, 0.5)) == p3
      assert p1.__radd__(p2) == p1.midpoint(p2).scale(2, 2)
      assert (-p3).__rsub__(p4) == p3.midpoint(p4).scale(2, 2)
  
      assert p4 * 5 == Point(5, 5)
      assert p4 / 5 == Point(0.2, 0.2)
      assert 5 * p4 == Point(5, 5)
  
      raises(ValueError, lambda: Point(0, 0) + 10)
  
      # Point differences should be simplified
      assert Point(x * (x - 1), y) - Point(x**2 - x, y + 1) == Point(
          0, -1
      )
  
      a, b = S.Half, Rational(1, 3)
      assert Point(a, b).evalf(2) == Point(
          a.n(2), b.n(2), evaluate=False
      )
      raises(ValueError, lambda: Point(1, 2) + 1)
  
      # test project
      assert Point.project((0, 1), (1, 0)) == Point(0, 0)
      assert Point.project((1, 1), (1, 0)) == Point(1, 0)
      raises(ValueError, lambda: Point.project(p1, Point(0, 0)))
  
      # test transformations
      p = Point(1, 0)
      assert p.rotate(pi / 2) == Point(0, 1)
      assert p.rotate(pi / 2, p) == p
      p = Point(1, 1)
      assert p.scale(2, 3) == Point(2, 3)
      assert p.translate(1, 2) == Point(2, 3)
      assert p.translate(1) == Point(2, 1)
      assert p.translate(y=1) == Point(1, 2)
      assert p.translate(*p.args) == Point(2, 2)
  
      # Check invalid input for transform
      raises(ValueError, lambda: p3.transform(p3))
      raises(ValueError, lambda: p.transform(Matrix([[1, 0], [0, 1]])))
  
      # test __contains__
      assert 0 in Point(0, 0, 0, 0)
      assert 1 not in Point(0, 0, 0, 0)
  
      # test affine_rank
      assert Point.affine_rank() == -1
  
imports: from sympy.core.basic import Basic
  from sympy.core.numbers import I, Rational, pi
  from sympy.core.parameters import evaluate
  from sympy.core.singleton import S
  from sympy.core.symbol import Symbol
  from sympy.core.sympify import sympify
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.geometry import Line, Point, Point2D, Point3D, Line3D, Plane
  from sympy.geometry.entity import rotate, scale, translate, GeometryEntity
  from sympy.matrices import Matrix
  from sympy.utilities.iterables import subsets, permutations, cartes
  from sympy.utilities.misc import Undecidable
  from sympy.testing.pytest import raises, warns
label_desc: Use image node_id 1 for calling a global function with example usage: test_point() without return types