Compare d.S and Heron's formula for face area

face-area.py

#!/usr/bin/env python3
import math
import numpy as np
import numpy.linalg as la

def herons_area(v1, v2, v3):
    a = la.norm(v2 - v1)
    b = la.norm(v3 - v2)
    c = la.norm(v1 - v3)
    p = 0.5*(a+b+c)
    return math.sqrt(p*(p-a)*(p-b)*(p-c))

x1 = 1
y1 = 1

x2 = 3
y2 = 0.5

x3 = x1 # assume this is always true!
y3 = 6

v0 = np.array([0, 0])
v1 = np.array([x1, y1])
v2 = np.array([x2, y2])
v3 = np.array([x3, y3])

area1 = herons_area(v0, v1, v3)
area2 = herons_area(v1, v2, v3)
print("area", area1+area2)

d = v2
S = np.array([y3 - y1, 0]) # should really use cross product                                                

print("0.5*S.d", 0.5*np.dot(S,d))