diff --git a/ch7/polygon.py b/ch7/polygon.py
index 54cf62f..2db6ddd 100644
--- a/ch7/polygon.py
+++ b/ch7/polygon.py
@@ -16,10 +16,7 @@ def __init__(self, x, y):                       # constructor
     self.x = x
     self.y = y
   
-  def __sub__(self, b): #tovec(a,b)==b-a
-    return point(self.x-b.x, self.y-b.y)
-  def __lt__(self, b):  #self<b mod for floats :)
-    return (self.x<b.x) if self.x!=b.x else (self.y<b.y)
+  def __lt__(self, b): return (self.x, self.y) < (b.x, b.y)
   
   def __str__(self): return "{} {}".format(self.x, self.y)
   def __hash__(self):return hash((self.x,self.y))
@@ -28,6 +25,13 @@ def __hash__(self):return hash((self.x,self.y))
 #   bool operator <(const point &p) const {
 #    return x < p.x || (abs(x-p.x) < EPS && y < p.y); } };
 
+class vec:
+  def __init__(self, x=0, y=0):
+    self.x = x
+    self.y = y
+
+def toVec(a, b):
+  return vec(b.x-a.x, b.y-a.y)
 # struct vec { double x, y;  // name: `vec' is different from STL vector
 #   vec(double _x, double _y) : x(_x), y(_y) {} };
 
@@ -39,12 +43,15 @@ def dist(p1, p2):                                 # Euclidean distance
 
 # returns the perimeter of polygon P, which is the sum of
 # Euclidian distances of consecutive line segments (polygon edges)
+# def perimeter(P) return math.fsum(dist(P[i], P[i+1]) for i in range(len(P)-1)) 
 def perimeter(P):
   ans = 0.0
   for i in range(len(P)-1):                       # note: P[n-1] = P[0]
     ans += dist(P[i], P[i+1])                     # as we duplicate P[0]
   return ans
 
+def area(P):
+  return math.fsum(cross(P[i], P[i+1]) for i in range(len(P)-1))/2
 # // returns the area of polygon P
 # double area(const vector<point> &P) {
 #   double ans = 0.0;
@@ -73,7 +80,7 @@ def cross(a, b): return a.x*b.y-a.y*b.x
 #   return fabs(ans)/2.0;
 # }
 
-def ccw(p, q, r): return (cross(q-p, r-p) > 0)
+def ccw(p, q, r): return (cross(toVec(p,q),toVec(p,r)) > 0)
 # note python i used class opperators for tovec (Agis Daniels)
 # // note: to accept collinear points, we have to change the `> 0'
 # // returns true if point r is on the left side of line pq
@@ -181,8 +188,8 @@ def CH_Andrew(ps):
   def f(B): #f is a mapping of the two loops since its dup code
     for p in P:
       while len(H)>B and not ccw(H[-2], H[-1], p): H.pop()
-        H.append(p)
-      H.pop()
+      H.append(p)
+    H.pop()
   f(1); P=P[::-1]; f(len(H)+1); return H #4 line low, rev, up, ret
 #c++ implementation below 
 # vector<point> CH_Andrew(vector<point> &Pts) {    // overall O(n log n)