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 <cstdlib>
#include <cmath>
#include <climits>
#include <cfloat>
#include <map>
//#include <utility>
#include <set>
#include <iostream>
//#include <memory>
#include <string>
#include <vector>
#include <algorithm>
//#include <functional>
#include <sstream>
//#include <deque>
#include <complex>
#include <stack>
#include <queue>
#include <cstdio>
//#include <cctype>
#include <cstring>
//#include <ctime>
#include <iterator>
#include <bitset>
//#include <numeric>
//#include <list>
//#include <iomanip>

#if __cplusplus >= 201103L
#include <array>
#include <tuple>
#include <initializer_list>
#include <unordered_set>
#include <unordered_map>
#include <forward_list>

#define cauto const auto&
#else

#endif

using namespace std;


namespace{
typedef long long ll;
typedef pair<int,int> pii;
typedef pair<ll,ll> pll;

typedef vector<int> vint;
typedef vector<vector<int> > vvint;
typedef vector<long long> vll, vLL;
typedef vector<vector<long long> > vvll, vvLL;

#define VV(T) vector<vector< T > >

template <class T>
void initvv(vector<vector<T> > &v, int a, int b, const T &t = T()){
    v.assign(a, vector<T>(b, t));
}

template <class F, class T>
void convert(const F &f, T &t){
    stringstream ss;
    ss << f;
    ss >> t;
}


#define reep(i,a,b) for(int i=(a);i<(b);++i)
#define rep(i,n) reep((i),0,(n))
#define ALL(v) (v).begin(),(v).end()
#define PB push_back
#define F first
#define S second
#define mkp make_pair
#define RALL(v) (v).rbegin(),(v).rend()


#define MOD 1000000007LL
#define EPS 1e-8
static const int INF=1<<24;


typedef complex<double> P;
typedef const P &rP;
typedef pair<P,P> seg;
//naiseki
double dot(rP a, rP b){
    return real(a) * real(b) + imag(a) * imag(b);
}
//gaiseki
double cross(rP a, rP b){
    return real(a) * imag(b) - imag(a) * real(b);
}
bool intersectSP(P a,P b, const P &p) {
  return abs(a-p)+abs(b-p)-abs(b-a) < EPS; // triangle inequality
}
P projection(P a,P b, const P &p) {
  double t = dot(p-a, a-b) / norm(a-b);
  return a + t*(a-b);
}
double distanceSP(P a,P b, const P &p) {
  const P r = projection(a,b, p);
  if (intersectSP(a,b, r)) return abs(r - p);
  return min(abs(a - p), abs(b - p));
}
bool foo(P a,P b,P c,P d){
	P aa=a-c,ar=d-a;
	P bb=b-a,br=d-b;
	P cc=c-b,cr=d-c;
	double a2=dot(aa,ar),b2=dot(bb,br),c2=dot(cc,cr);
	if(a2>EPS&&b2>EPS&&c2>EPS) return true;
	if(a2<-EPS&&b2<-EPS&&c2<-EPS) return true;
	return false;
}
void mainmain(){
	double r;
	double x,y;
	while(cin>>x>>y,x||y){
		// cout<<x<<" "<<y<<endl;
		vector<P> z;
		z.PB(P(x,y));
		rep(i,3){
			cin>>x>>y;
			z.PB(P(x,y));
		}
		cin>>r;
		// cout<<r<<endl;
		if(r>=abs(z[0]-z[3])&&r>=abs(z[1]-z[3])&&r>=abs(z[2]-z[3])){
			cout<<"b\n";
			continue;
		}
		bool f=true;
		rep(i,3){
			double t=distanceSP(z[i%3],z[(i+1)%3],z[3]);
			// cout<<t<<endl;
			if(t<r){
				// cout<<real(z[i%3])<<" "<<imag(z[(i+1)%3])<<endl;
				f=false;
			}
		}
		if(f&&foo(z[0],z[1],z[2],z[3])){
			cout<<"a\n";
		}
		else if(r>=abs(z[0]-z[3])||r>=abs(z[1]-z[3])||r>=abs(z[2]-z[3])){
			cout<<"c\n";
		}
		else cout<<"d\n";
		// cout<<"aaa"<<endl;	
	}
}



}
main() try{
    mainmain();
}
catch(...){
}

C++: #include<iostream>
#include<cstdlib>
#include<complex>
#define EPS 1e-10
using namespace std;

typedef complex<double> Po;
Po p[3],o;

double dot(Po x, Po y){return x.real()*y.real()+x.imag()*y.imag();}

double cross(Po x, Po y){return x.real()*y.imag() - x.imag()*y.real();}

double area(Po a, Po b, Po c){return abs(cross(b-a,c-a))/2;}

bool inter(Po a, Po b, Po c, Po x){
  return (fabs(area(a,b,x)+area(b,c,x)+area(c,a,x)-area(a,b,c)) < EPS);
}

double seg_dis(Po a, Po b, Po 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);
}

int main(){
  double x[4],y[4],r;

  for(;;){
    cin >> x[0] >> y[0];
    if(!x[0] && !y[0])break;

    for(int i=1;i<4;i++)cin >> x[i] >> y[i];
    cin >> r;

    for(int i=0;i<3;i++)p[i] = Po(x[i],y[i]);
    o = Po(x[3],y[3]);


    bool f = true;;
    for(int i=0;i<3;i++)f &= !(abs(o-p[i])>r);
    if(f)cout << "b\n";
    else if(inter(p[0],p[1],p[2],o)){
      f = true;
      for(int i=0;i<3;i++)f &= !(seg_dis(p[i],p[(i+1)%3],o)<r);
      if(f)cout << "a\n";
      else cout << "c\n";
    }else{
      for(int i=0;i<3;i++)f |= !(seg_dis(p[i],p[(i+1)%3],o)>r);
      if(f)cout << "c\n";
      else cout << "d\n";
    }
  }
}