Code to work out the area of the quadrilateral defined by joining each vertex to the next-but-one midpoint of an edge, where the original quadrilateral has vertices (0,0), (2,0), (a,b), and (0,2).

central_quad.py

def AC(a,b):
  return [ 2*a/(a+2*b+4), (2*b+4)/(a+2*b+4)]
def AD(a,b):
  return [ (2*a+2*b-2)/(a+2*b-1), b/(a+2*b-1)]
def BD(a,b):
  return [ (a+2)*(2*a+b-2)/(4*a+3*b-4), b*(2*a+b)/(4*a+3*b-4)]
def BC(a,b):
  return [ a*(a+2)/(3*a+b+2), (a+2)*(b+2)/(3*a+b+2) ]
def Area(a,b):
  ac = AC(a,b)
  ad = AD(a,b)
  bd = BD(a,b)
  bc = BC(a,b)
  return (ac[0] * ad[1] + ad[0] * bd[1] + bd[0] * bc[1] + bc[0] * ac[1] - ac[1] * ad[0] - ad[1] * bd[0] - bd[1] * bc[0] - bc[1] * ac[0])/2