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 <iostream>
#include <sstream>
#include <string>
#include <algorithm>
#include <vector>
#include <stack>
#include <queue>
#include <complex>
#include <set>
#include <map>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <cmath>
#include <cassert>

using namespace std;

#define FOR(i,k,n) for(int i=(k); i<(int)(n); ++i)
#define REP(i,n) FOR(i,0,n)
#define FORIT(i,c) for(__typeof((c).begin())i=(c).begin();i!=(c).end();++i)

template<class T> void debug(T begin, T end){ for(T i = begin; i != end; ++i) cerr<<*i<<" "; cerr<<endl; }
inline bool valid(int x, int y, int W, int H){ return (x >= 0 && y >= 0 && x < W && y < H); }

typedef long long ll;
const int INF = 100000000;
const double EPS = 1e-8;
const int MOD = 1000000007;
int dx[8] = {1, 0, -1, 0, 1, -1, -1, 1};
int dy[8] = {0, 1, 0, -1, 1, 1, -1, -1};

typedef complex<double> P;
typedef pair<P, P> L;
#define X real()
#define Y imag()

double dot(P& a, P& b){
  return a.X * b.X + a.Y * b.Y;
}
double cross(P& a, P& b){
  return a.X * b.Y - a.Y * b.X;
}

int sign(double x){
  if(x > EPS) return 1;
  if(x < -EPS) return -1;
  return 0;
}

//a -> b, a -> c
int ccw(P a, P b, P c){
  b -= a; c -= a;
  if(sign(cross(b, c)) > 0) return 1; //反時計
  if(sign(cross(b, c)) < 0) return -1; //時計
  if(sign(dot(b, c)) < 0) return +2; //c-a-b
  if(norm(b) < norm(c)) return -2; //a-b-c
  return 0; //a-c-b
}

double min_d(P a, P b, P c){
  double d1 = abs(c - a);
  double d2 = abs(c - b);
  b -= a; c -= a;
  double k = dot(b, c) / norm(b);
  if(0 < k && k < 1) return abs(c - k * b);
  else return min(d1, d2);
}

string solve(P p[3], P c, double r){
  if(ccw(p[0], p[1], p[2]) == -1) swap(p[1], p[2]);
  double maxdist = 0, mindist = 1e18;
  int ccw_v[3];
  REP(i, 3){
    maxdist = max(maxdist, abs(p[i] - c));
    mindist = min(mindist, min_d(p[i], p[(i + 1) % 3], c));
    ccw_v[i] = ccw(p[i], p[(i + 1) % 3], c);
  }
  if(sign(maxdist - r) <= 0) return "b";
  if(ccw_v[0] == ccw_v[1] && ccw_v[1] == ccw_v[2] && ccw_v[0] == 1){
    if(sign(mindist - r) >= 0) return "a";
    else return "c";
  }else{
    if(sign(mindist - r) > 0) return "d";
    else return "c";
  }
}

int main(){
  P p[3];
  P c; double r;
  while(cin>>p[0].X>>p[0].Y>>p[1].X>>p[1].Y>>p[2].X>>p[2].Y>>c.X>>c.Y>>r){
    cout<<solve(p, c, r)<<endl;
  }
  return 0;
}

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

typedef complex<double> Po;


Po p[3],o;

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

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;
  int i;

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

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

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


    bool f = true;;
    for(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(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(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";
    }
  }
}