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 <complex>
#include <sstream>
#include <string>
#include <algorithm>
#include <deque>
#include <list>
#include <map>
#include <numeric>
#include <queue>
#include <vector>
#include <set>
#include <limits>
#include <cstdio>
#include <cctype>
#include <cmath>
#include <cstring>
#include <cstdlib>
#include <ctime>
using namespace std;
#define REP(i, j) for(int i = 0; i < (int)(j); ++i)
#define FOR(i, j, k) for(int i = (int)(j); i < (int)(k); ++i)
#define SORT(v) sort((v).begin(), (v).end())
#define REVERSE(v) reverse((v).begin(), (v).end())
#define curr(P, i) P[i]
#define next(P, i) P[(i+1)%P.size()]
const double EPS = 1e-8;
const int N = 3;
typedef complex<double> P;

typedef vector<P> G;

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

double cross(const P& a, const P& b) { return imag(conj(a)*b); }
double dot(const P& a, const P& b) { return real(conj(a)*b); }

bool intersectSP(const L &s, const P &p) {
  return abs(s[0]-p)+abs(s[1]-p)-abs(s[1]-s[0]) < EPS; // triangle inequality
}

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 distanceSP(const L &s, const P &p) {
  const P r = projection(s, p);
  if (intersectSP(s, r)) return abs(r - p);
  return min(abs(s[0] - p), abs(s[1] - p));
}

enum { OUT, ON, IN };
int contains(const G &g, const P &p) {
  bool in = false;
  for (int i = 0; i < g.size(); ++i) {
    P a = curr(g,i) - p, b = next(g,i) - p;
    if (imag(a) > imag(b)) swap(a, b);
    if (imag(a) <= 0 && 0 < imag(b))
      if (cross(a, b) < 0) in = !in;
    if (cross(a, b) == 0 && dot(a, b) <= 0) return ON;
  }
  return in ? IN : OUT;
}

bool input(P &p){
  int x, y; cin >>x >>y;
  p = P(x, y);
  return !(x == 0 && y == 0);
}

char solve(G &g, P &p, int r){
  double maxd = 0, mind = 1e9;
  REP(i, N){
    maxd = max(maxd, abs(curr(g, i) - p));
    mind = min(mind, distanceSP(L(curr(g, i), next(g, i)), p));
  }
  if(maxd <= r) return 'b';
  int cont = contains(g, p);
  if(cont == IN && mind >= r) return 'a';
  if(cont == OUT && mind > r) return 'd';
  return 'c';
}

int main() {
  G g(N);
  while(input(g[0])){
    FOR(i, 1, N) input(g[i]);
    P p; int r;
    input(p); cin >>r;
    cout <<solve(g, p, 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 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[3];
    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{
      if(inter(p[0],p[1],p[2],o)){
	for(int i=0;i<3;i++)f[i] = (seg_dis(p[i],p[(i+1)%3],o)>r+EPS);
	if(f[0] && f[1] && f[2])cout << "a\n";
	else cout << "c\n";
      }else{
	for(int i=0;i<3;i++)f[i] = (seg_dis(p[i],p[(i+1)%3],o)<r+EPS);
	if(f[0] || f[1] || f[2])cout << "c\n";
	else cout << "d\n";
      }
    }
  }
}