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>

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;
}

vector<segment> construct_edge(point* p, int n){
  vector<segment>seg,seg2;
  segment temp;
  for(int i=0;i<n;i++){
    for(int j=0;j<n;j++){
      if(i!=j){
	temp.a=p[i];
	temp.b=p[j];
	seg.push_back(temp);
      }
    }
  }

  int t=seg.size();
  for(int i=0;i<t;i++){
    int fg=0;
    for(int j=0;j<t;j++){
      if(i!=j && seg[i].b.x==seg[j].a.x && seg[i].b.y==seg[j].a.y){
	if(ccw(seg[i].a,seg[i].b,seg[j].b)==1){
	  fg=1;
	  break;
	}
      }
    }
    if(!fg)seg2.push_back(seg[i]);
  }
  return seg2;
}

int point_contain(vector<segment> ps, point p, int n){
  int cnt=ccw(ps[0].a,ps[0].b,p);

  for(int i=0;i<n;i++)
    if(cnt!=ccw(ps[i].a,ps[i].b,p))return 0;
  
  return 1;
}

int containCS(point a,point b,circle c){
  if(distance_ls_p(a,b,c.p)<c.r+EPS)return 1;
  else return 0;
}

int main(void){
  point p[3];
  vector<segment> t;
  circle C;

  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;

    double ans=0;
    for(int i=0;i<3;i++){
	int tmp=dist(p[i],C.p);
	if(tmp>ans){
	  ans=tmp;
	}
    }
    if(C.r>ans){
      cout <<'b'<<endl;
      continue;
    }
    
    t=construct_edge(p,3);
    int res1=point_contain(t,C.p,3);
    double res2=100000000;
    for(int i=0;i<3;i++)
      res2=min(res2,distance_ls_p(t[i].a,t[i].b,C.p));

    if(res1 && C.r<=res2)cout <<'a'<< endl;
    else if(!res1 && C.r<res2)cout <<'d'<< endl;
    else cout <<'c'<< 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 b[3]){
	P a[3];
	for(int i = 0 ; i < 3 ; i++) a[i] = b[i];
	a[0] -= a[2];
	a[1] -= a[2];
	return 0.5 * abs( a[0].X * a[1].Y - a[1].X * a[0].Y );
}


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);
		}
		//cout << S << endl;
		bool in = S == 0;
		bool hit2 = getPedal(L(a[0],a[1]),c) <= r || getPedal(L(a[1],a[2]),c) <= r || getPedal(L(a[0],a[2]),c) <= r;
		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){
			if(hit) cout << "c" << endl;
			else cout << "a" << endl;
		}else{
			if(hit2) cout << "c" << endl;
			else cout << "d" << endl;
		}
	}
}