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>
#include<climits>

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;
}

bool cmp_x(const point& p, const point& q){
  if(p.x != q.x)return p.x<q.x;
  return p.y < q.y;
}

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 is_intersected_ls(point a1, point a2, point b1, point b2) {
  return ( cross(a2-a1, b1-a1) * cross(a2-a1, b2-a1) < EPS ) &&
         ( cross(b2-b1, a1-b1) * cross(b2-b1, a2-b1) < EPS );
}

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;
}

int inside(point t,point* p,int N){
  int i,count=0,j=0;
  segment lt,lp;
  sort(p,p+N,cmp_x);
  p[N]=p[0];
  p[N+1]=p[1];

  lt.a=lt.b=t;
  lt.b.x=INT_MAX;

  for(i=1;i<=N;i++){
    lp.a=lp.b=p[i];

    if(!is_intersected_ls(lp.a,lp.b,lt.a,lt.b)){
      if(i==j+1){
	lp.b=p[j];
	if(is_intersected_ls(lp.a,lp.b,lt.a,lt.b))count++;
      }
      else if(ccw(lt.a,lt.b,p[i])*ccw(lt.a,lt.b,p[j])<0)count++;
      j=i;
    }
  }
  return count & 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;

  int res1=inside(C.p,ps,sz);

  double res2=100000000;
  ps[3]=ps[0];
  for(int i=0;i<sz;i++)
    res2=min(res2,distance_ls_p(ps[i],ps[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[6];
  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 <complex>
#include <vector>
#include <algorithm>
using namespace std;

typedef complex<double> P;

#define EPS (1e-10)

double dot(P a,P b){
	return (a.real() * b.real() + a.imag() * b.imag());
}

double cross(P a,P b){
	return (a.real() * b.imag() - a.imag() * b.real());
}

double distance_ls_p(P a,P b,P c){
	if(dot(b-a,c-a) < EPS) return abs(c-a);
	if(dot(a-b,c-b) < EPS) return abs(c-b);
	return abs(cross(b-a,c-a)) / abs(b-a);
}

bool pol_contains_cir(vector<P> p,P o,double r){
	int n = p.size();
	int cr = 0, cl = 0;
	for(int i=0;i<n;i++){
		P a = p[i];
		P b = p[(i+1)%n];
		if(EPS > cross(b-a,o-a)) cr++;
		if(0 < cross(b-a,o-a) + EPS) cl++;
		if(distance_ls_p(a,b,o) < r) return false;
	}
	if(cr != n && cl != n) return false;

	return true;
}

bool cir_contains_pol(vector<P> p,P o,double r){
	for(int i=0;i<p.size();i++){
		if(r < abs(p[i]-o)) return false;
	}
	return true;
}

bool is_intersected_ls_cir(P a,P b,P o,double r){
	double ao = abs(a-o);
	double bo = abs(b-o);

	return ((ao < r + EPS && r < bo + EPS || r < ao + EPS && bo < r + EPS) ||
					(r < ao + EPS && r < bo + EPS && distance_ls_p(a,b,o) < r + EPS));
}

bool is_intersected_pol_cir(vector<P> p,P o,double r){
	int n = p.size();

	for(int i=0;i<n;i++){
		if(is_intersected_ls_cir(p[i],p[(i+1)%n],o,r)) return true;
	}

	return false;
}

int main(void){
	/*
	P a = P(0,0);
	P b = P(0,2);
	P c = P(-0.1,1);

	cout << cross(b-a,c-a) << endl;
	*/

	while(1){
		vector<P> p;
		for(int i=0;i<3;i++){
			P tmp;
			cin>>tmp.real()>>tmp.imag();
			if(tmp.real()==0 && tmp.imag()==0) return 0;
			p.push_back(tmp);
		}

		P o;
		double r;

		cin>>o.real()>>o.imag()>>r;

		if(pol_contains_cir(p,o,r))
			cout<<"a\n";
		else if(cir_contains_pol(p,o,r)){
			cout<<"b\n";
		}
		else if(is_intersected_pol_cir(p,o,r)){
			cout<<"c\n";
		}
		else{
			cout<<"d\n";
		}
	}
}