Given four points in a plane, defined by their coordinates, there are 4!=24 permutations of those points. The scripts computes the area of the 24 different quadrilaterals with the shoelace formula and plot them. The maximal area found, seems to correspond to non self crossing direct quadrilaterals.

QuadrilateralArea.py

# -*- coding: utf-8 -*-
"""
Created on Tue Dec 18 08:33:15 2012
Modif 3 january 2013 to use matplotlib.patches.Polygon

@author: Jean-Patrick Pommier

Test the shoelace formula to compute the area of a quadrilateral:
    . Chose four points
    . test if self intersection occurs:
        http://www.bryceboe.com/2006/10/23/line-segment-intersection-algorithm/
    . compute area:
        http://en.wikipedia.org/wiki/Shoelace_formula
"""

import itertools as it
from matplotlib import pyplot as plt
from matplotlib.patches import Polygon


def quadAreaShoelace(A,B,C,D):
    x1 = A[0]
    y1 = A[1]
    x2 = B[0]
    y2 = B[1]
    x3 = C[0]
    y3 = C[1]
    x4 = D[0]
    y4 = D[1]
    #sign +/- if direct/indirect quadrilateral
    return 0.5*(x1*y2+x2*y3+x3*y4+x4*y1-x2*y1-x3*y2-x4*y3-x1*y4)

def ccw(A,B,C):
    return (C[1]-A[1]) * (B[0]-A[0]) > (B[1]-A[1]) * (C[0]-A[0])


def intersect(A,B,C,D):
    # Return true if line segments AB and CD intersect
    return ccw(A,C,D) != ccw(B,C,D) and ccw(A,B,C) != ccw(A,B,D)

A = (1.1,0.3)
B = (1.7,1.4)
C = (1,2) 
D = (0,1)

n = 1
for quad in it.permutations((A,B,C,D)):
    area = quadAreaShoelace(quad[0],quad[1],quad[2],quad[3])
    print quad, intersect(quad[0],quad[1],quad[2],quad[3]), area
    plt.subplot(4,6,n,frameon = True, xticks = [], yticks = [])#xticks = [], yticks = []
    plt.xlim((0,2))
    plt.ylim((0,2))
    startpoint=quad[0]
    plt.scatter(startpoint[0],startpoint[1],c='m',s=100)
    for i in range(len(quad)):
        point1 = quad[i]
        point2 = quad[(i+1)%4]
        x = point1[0]
        y = point1[1]
        nextpoint = quad[(i+1)%4]
        dx = nextpoint[0]-x
        dy = nextpoint[1]-y

        if area<0:
            Color='blue'
        if area == 0:
            Color='black'
        if area > 0:
            Color = 'red'    
        plt.arrow(x,y,dx,dy, color=Color,shape='full', lw=1, length_includes_head=True, head_width=.2)
    p = Polygon( quad, alpha=0.2, color='g' )
    plt.gca().add_artist(p)
    n = n+1
plt.show()