def prog ( ):

  import matplotlib.pyplot as plt
  import numpy as np

  t = np.array ( [ \
    [ 4, 1 ], \
    [ 8, 3 ], \
    [ 0, 9 ] ] )

  print ( t )

  centroid = np.mean ( t, axis = 0 )
  print ( centroid )

  print ( triangle_sides ( t ) )

  angles = triangle_angles ( t )
  print ( angles )
  print ( angles * 180 / np.pi )

  area = triangle_area ( t )
  print ( area )

  area = triangle_area_signed ( t )
  print ( area )

  t1 = np.array ( [ \
    [ 4, 1 ], \
    [ 0, 9 ], \
    [ 8, 3 ] ] )

  area = triangle_area_signed ( t1 )
  print ( area )

  print ( triangle_orientation ( t ) )
  print ( triangle_orientation ( t1 ) )

  print ( triangle_contains_point ( t, [ 1, 2 ] ) )
  print ( triangle_contains_point ( t, [ 2, 5 ] ) )
  print ( triangle_contains_point ( t, [ 4, 4 ] ) )

  return

def line_side ( p0, p1, q ):
  import numpy as np
  v0 = p1 - p0
  v1 = q -  p0
  v0xv1 = float ( np.cross ( v0, v1 ) )
  return v0xv1

def triangle_angles ( t ):

  import numpy as np

  s = triangle_sides ( t )

  angle = np.zeros ( 3 )
  for i in range ( 0, 3 ):
    ip1 = ( i + 1 ) % 3
    ip2 = ( i + 2 ) % 3
    top = s[i]**2 + s[ip2]**2 - s[ip1]**2
    bot = 2.0 * s[i] * s[ip2]
    angle[i] = np.arccos ( top / bot )

  return angle

def triangle_area ( t ):

  import numpy as np

  s = triangle_sides ( t )
  abc = 0.5 * np.sum ( s )
  area = np.sqrt ( abc * ( abc - s[0] ) * ( abc - s[1] ) * ( abc - s[2] ) )

  return area

def triangle_area_signed ( t ):

  import numpy as np

  area = 0.5 * np.cross ( ( t[1,:] - t[0,:] ), ( t[2,:] - t[0,:] ) )

  return area

def triangle_contains_point ( t, q ):

  value = True
  for i in range ( 0, 3 ):
    ip1 = ( i + 1 ) % 3
    if ( line_side ( t[i,:], t[ip1,:], q ) < 0.0 ):
      value = False
      return value

  return value

def triangle_orientation ( t ):

  import numpy as np

  area = triangle_area_signed ( t )
  if ( 0 <= area ):
    orientation = +1
  else:
    orientation = -1

  return orientation

def triangle_sides ( t ):

  import numpy as np

  s = np.zeros ( 3 )
  for i in range ( 0, 3 ):
    ip1 = ( i + 1 ) % 3
    ip2 = ( i + 2 ) % 3
    s[i] = np.linalg.norm ( t[ip2,:] - t[ip1,:] )

  return s


prog ( )