Go: package main

import (
	"bufio"
	"fmt"
	"math"
	"os"
	"strconv"
	"strings"
)

var scanner = bufio.NewScanner(os.Stdin)

func main() {
	for scanner.Scan() {
		buf0 := strings.Split(scanner.Text(), " ")
		input0 := make([]int, 2)
		for i := range input0 {
			input0[i], _ = strconv.Atoi(buf0[i])
		}
		if input0[0] == 0 && input0[1] == 0 {
			break
		}
		a, b := input0[0], input0[1]
		scanner.Scan()
		buf1 := strings.Split(scanner.Text(), " ")
		input1 := make([]int, 2)
		for i := range input1 {
			input1[i], _ = strconv.Atoi(buf1[i])
		}
		c, d := input1[0], input1[1]
		scanner.Scan()
		buf2 := strings.Split(scanner.Text(), " ")
		input2 := make([]int, 2)
		for i := range input2 {
			input2[i], _ = strconv.Atoi(buf2[i])
		}
		e, f := input2[0], input2[1]
		scanner.Scan()
		buf3 := strings.Split(scanner.Text(), " ")
		input3 := make([]int, 2)
		for i := range input3 {
			input3[i], _ = strconv.Atoi(buf3[i])
		}
		g, h := input3[0], input3[1]
		scanner.Scan()
		r, _ := strconv.Atoi(scanner.Text())
		if (a-g)*(a-g)+(b-h)*(b-h) <= r*r && (c-g)*(c-g)+(d-h)*(d-h) <= r*r && (e-g)*(e-g)+(f-h)*(f-h) <= r*r {
			fmt.Println("b")
		} else if dist(a, b, c, d, g, h) > float64(r)-0.000001 && dist(a, b, e, f, g, h) > float64(r)-0.000001 && dist(e, f, c, d, g, h) > float64(r)-0.000001 && inclusive(a, b, c, d, e, f, g, h) {
			fmt.Println("a")
		} else if CL_intersect(a, b, c, d, g, h, r) || CL_intersect(a, b, e, f, g, h, r) || CL_intersect(e, f, c, d, g, h, r) {
			fmt.Println("c")
		} else {
			fmt.Println("d")
		}
	}
}

func CL_intersect(ax, ay, bx, by, cx, cy, r int) bool {
	acx := cx - ax
	acy := cy - ay
	bcx := cx - bx
	bcy := cy - by
	abx := bx - ax
	aby := by - ay
	if acx*acx+acy*acy >= r*r || bcx*bcx+bcy*bcy >= r*r {
		if abx*acx+aby*acy < 0 {
			if acx*acx+acy*acy <= r*r {
				return true
			}
		} else {
			if abx*acx+aby*acy > abx*abx+aby*aby {
				if bcx*bcx+bcy*bcy <= r*r {
					return true
				}
			} else if acx*acx+acy*acy-(abx*acx+aby*acy)*(abx*acx+aby*acy)/(abx*abx+aby*aby) <= r*r {
				return true
			}
		}
	}
	return false
}

func inclusive(ax, ay, bx, by, cx, cy, px, py int) bool {
	ok1 := true
	ok2 := true
	if (bx-ax)*(py-ay)-(px-ax)*(by-ay) < 0 {
		ok1 = false
	}
	if (bx-ax)*(py-ay)-(px-ax)*(by-ay) > 0 {
		ok2 = false
	}
	if (cx-bx)*(py-by)-(px-bx)*(cy-by) < 0 {
		ok1 = false
	}
	if (cx-bx)*(py-by)-(px-bx)*(cy-by) > 0 {
		ok2 = false
	}
	if (ax-cx)*(py-cy)-(px-cx)*(ay-cy) < 0 {
		ok1 = false
	}
	if (ax-cx)*(py-cy)-(px-cx)*(ay-cy) > 0 {
		ok2 = false
	}
	return ok1 || ok2
}

func dist(ax, ay, bx, by, px, py int) float64 {
	dx := float64(bx - ax)
	dy := float64(by - ay)
	A := float64(dx*dx + dy*dy)
	B := dx*float64(ax-px) + dy*float64(ay-py)
	t := -B / A
	if t < 0 {
		t = 0
	}
	if t > 1 {
		t = 1
	}
	tx := float64(ax) + dx*t
	ty := float64(ay) + dy*t
	return math.Sqrt((float64(px)-tx)*(float64(px)-tx) + (float64(py)-ty)*(float64(py)-ty))
}

C++: #include<cmath>
#include<algorithm>
#include<iostream>
#include<vector>
#include<string>
 
using namespace std;
 
double EPS = 1e-10;
 
double add(double a, double b){
  if(abs(a+b) < EPS * (abs(a)+abs(b)))return 0;
  return a+b;
}
 
struct point{
  double x, y;
  point(){}
  point(double x,double y) : x(x) , y(y){}
 
  point operator + (point p){
    return point(add(x,p.x), add(y,p.y));
  }
 
  point operator - (point p){
    return point(add(x,-p.x), add(y,-p.y));
  }
 
  point operator * (double d){
    return point(x*d,y*d);
  }
 
  point operator / (double d){
    return point(x/d,y/d);
  }
};
 
struct segment{ point a,b; };
 
struct circle{ point p; double r; };
 
double dot(point a, point b) { return (a.x * b.x + a.y * b.y); }
 
double cross(point a, point b) { return (a.x * b.y - a.y * b.x); }
 
int ccw(point p0, point p1, point p2){
  point a = p1 - p0;
  point b = p2 - p0;
  if(cross(a,b)>EPS)return +1;
  if(cross(a,b)<EPS)return -1;
  if(dot(a,b)<-EPS)return +2;
  if(a.x*a.x+a.y*a.y<b.x*b.x+b.y*b.y)return -2;
  return 0;
}
 
double vec_abs(point a){
  return sqrt(a.x*a.x+a.y*a.y);
}
 
double distance_ls_p(point a, point b, point c) {
  if ( dot(b-a, c-a) < EPS ) return vec_abs(c-a);
  if ( dot(a-b, c-b) < EPS ) return vec_abs(c-b);
  return abs(cross(b-a, c-a)) / vec_abs(b-a);
}
 
double dist(point a,point b){
  return sqrt(pow(a.x-b.x,2)+pow(a.y-b.y,2));
}
 
int crossCS(point a, point b, circle c){
  double res=distance_ls_p(a,b,c.p);
  double ac=dist(a,c.p);
  double bc=dist(b,c.p);
 
  if(res>c.r || (ac<c.r && bc<c.r))return 0;
  else if(c.r-res<EPS ||(ac<c.r && bc>=c.r)||(ac>=c.r && bc<c.r))return 1;
  else return 2;
}
 
 bool cmp_x(const point& p, const point& q){
  if(p.x != q.x)return p.x<q.x;
  return p.y < q.y;
}

vector<point>convex_hull(point* ps, int n){
  sort(ps,ps+n,cmp_x);
  int k=0;
  vector<point>qs(n*2);
 
  for(int i=0;i<n;i++){
    while(k>1&&cross(qs[k-1]-qs[k-2],ps[i]-qs[k-1])<0)k--;
    qs[k++]=ps[i];
  }
 
  for(int i=n-2,t=k;i>=0;i--){
    while(k>t&&cross(qs[k-1]-qs[k-2],ps[i]-qs[k-1])<0)k--;
    qs[k++]=ps[i];
  }
  qs.resize(k-1);
  return qs;
}

int point_contain(point* ps, point p, int n){
  int cnt=ccw(ps[0],ps[1],p);
  ps[n++]=ps[0]; 

  for(int i=0;i<n-1;i++)
    if(cnt!=ccw(ps[i],ps[i+1],p))return 0;
   
  return 1;
}
 
int crossCP(point* ps, circle C, int sz){
 
  double ans=0;
 
  for(int i=0;i<sz;i++){
    double tmp=dist(ps[i],C.p);
    if(tmp>ans)ans=tmp;
  }
 
  if(C.r>=ans)return 2;
 
  vector<point>t=convex_hull(ps,sz);

  int tsz=t.size();
  int res1=point_contain(ps,C.p,tsz);
 
  double res2=100000000;
   t[t.size()]=t[0];
   tsz++;
  for(int i=0;i<tsz-1;i++)
    res2=min(res2,distance_ls_p(t[i],t[i+1],C.p));
 
  if(res1 && C.r<=res2)return 1;
  else if(!res1 && C.r<res2)return 4;
  else return 3;
}
 
int main(void){
  point p[3];
  vector<segment> t;
  circle C;
  string ans="zabcd";
 
  while(true){
    for(int i=0;i<3;i++){
      cin >> p[i].x >> p[i].y;
      if(p[i].x==0 && p[i].y==0)return 0;
    }
    cin >> C.p.x >> C.p.y >> C.r;
 
    int res=crossCP(p,C,3);
    cout << ans[res] << endl;
  }
  return 0;
}

C++: #include <iostream>
#include <cmath>
#include <algorithm>
#include <complex>
#include <vector>
using namespace std;
typedef long double Double;
typedef complex<Double> P;
#define Y imag()
#define X real()
const Double EPS = (long double) 1 * 1e-12;

struct L : public vector<P> {L(const P &a, const P &b) {push_back(a); push_back(b);} };

Double get(P a[3]){
	Double A = abs(a[0]-a[1]);
	Double B = abs(a[1]-a[2]);
	Double C = abs(a[2]-a[0]);
	Double s = (A+B+C) / 2.0;
	Double S = sqrt( s * (s-A) * (s-B) * (s-C) );
	return S;
}
Double nijo(P a){
	return a.X * a.X + a.Y * a.Y;
}
 
Double dot(const P& a, const P& b) {return real(conj(a)*b);}
P projection(const L &l, const P &p) {Double t = dot(p-l[0], l[0]-l[1]) / norm(l[0]-l[1]);return l[0] + t*(l[0]-l[1]);}

Double getPedal(const L &l, const P &p) {
	return abs(p - projection(l, p));
}

int main(){
	P a[3],c;Double r;
	while(cin >> a[0].X >> a[0].Y && (a[0].X != 0 || a[0].Y != 0)){
		for(int i = 1 ; i < 3 ; i++) cin >> a[i].X >> a[i].Y;
		cin >> c.X >> c.Y >> r;
		Double S = get(a);
		for(int i = 0 ; i < 3 ; i++){
			P b[3];
			b[0] = a[i];
			b[1] = a[(i+1)%3];
			b[2] = c;
			S -= get(b);
		}
		bool in = abs(S) < EPS;
		bool hit = getPedal(L(a[0],a[1]),c) <= r || getPedal(L(a[1],a[2]),c) <= r || getPedal(L(a[0],a[2]),c) <= r;
		//cout << getPedal(L(a[0],a[1]),c) << " " << r << endl;
		if(  nijo(a[0]-c) < r*r && nijo(a[1]-c) < r*r && nijo(a[2]-c) < r*r) cout << "b" << endl;
		else if(in && !hit) cout << "a" << endl;
		else if(hit) cout << "c" << endl;
		else cout << "d" << endl;
	}
}