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>
using namespace std;

#define EPS 1.e-10
typedef complex<double> C;

C v[4];
double r;

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

double ncross(C &a, C &b) {
  return a.real()*b.imag() - a.imag()*b.real();
}

int tri_in_check() {
  int c[3];
  C a;
  double r2 = r*r;
  for (int i=0; i<3; i++) {
    a = v[i] - v[3];
    if (norm(a) < r2) {
      c[i] = -1;
    } else if (norm(a) > r2){
      c[i] = 1;
    } else {
      c[i] = 0;
    }
  }
  if (c[0]>=0. && c[1]>=0. && c[2]>=0.) {
    return 1;
  } else if (c[0]<=0. && c[1]<=0. && c[2]<=0.) {
    return -1;
  }
  return 0;
}

bool center_in_check() {
  C a, b;
  bool f[3];
  for (int i=0; i<3; i++) {
    a = v[(i+1)%3] - v[i];
    b = v[3] - v[(i+1)%3];
    f[i] = (ncross(a,b) > 0);
  }
  return (f[0]==f[1] && f[0]==f[2]);
}

int intersect() {
  C a, b;
  double na, nb, dab, d, sd, s, t;
  bool contact = false;
  for (int i=0; i<3; i++) {
    a = v[(i+1)%3] - v[i];
    b = v[3] - v[i];
    na = norm(a);
    nb = norm(b);
    dab = dot(a,b);
    d = dab*dab - na*(nb-r*r);
    if (d <=EPS) {
      if (fabs(d) < EPS) {
        contact = true;
      }
      continue;
    }
    sd = sqrt(d);
    s = (dab+sd)/na;
    t = (dab-sd)/na;
    if ((s>EPS&&s<1.-EPS) || (t>EPS&&t<=1.-EPS)) {
      return 1;
    }
    if ((fabs(s)<EPS && fabs(s-1.)<EPS) || (fabs(t)<EPS && fabs(t-1.)<EPS)) {
      contact = true;
    }
  }
  return (contact ? 0 : -1);
}

int main() {
  int t;
  while (1) {
    cin >> v[0].real() >> v[0].imag();
    if (!v[0].real() && !v[0].imag()) break;
    for (int i=1; i<4; i++) {
      cin >> v[i].real() >> v[i].imag();
    }
    cin >> r;
    if (tri_in_check() == -1) {
      cout << "b" << endl;
    } else {
      t = intersect();
      if (t == 1) {
        cout << "c" << endl;
      } else if (center_in_check()) {
        cout << "a" << endl;
      } else if (t == 0) {
        cout << "c" << endl;
      }  else {
        cout << "d" << endl;
      }
    }
  }
  return 0;
}

C++: #include <iostream>
#include <complex>
using namespace std;

typedef complex<double> C;


C v[4];
double r;

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

double ncross(C &a, C &b) {
  return a.real()*b.imag() - a.imag()*b.real();
}

bool tri_in_check() {
  bool c[3];
  C a;
  double r2 = r*r;
  for (int i=0; i<3; i++) {
    a = v[i] - v[3];
    if (norm(a) < r2) {
      c[i] = true;
    } else {
      c[i] = false;
    }
  }
  return (c[0] && c[1] && c[2]);
}

bool center_in_check() {
  C a, b;
  bool f[3];
  for (int i=0; i<3; i++) {
    a = v[(i+1)%3] - v[i];
    b = v[3] - v[(i+1)%3];
    f[i] = (ncross(a,b) > 0);
  }
  return (f[0]==f[1] && f[0]==f[2]);
}

bool intersect() {
  C a, b;
  double na, nb, dab, d, sd, s, t;
  for (int i=0; i<3; i++) {
    a = v[(i+1)%3] - v[i];
    b = v[3] - v[i];
    na = norm(a);
    nb = norm(b);
    dab = dot(a,b);
    d = dab*dab - na*(nb-r*r);
    if (d < 0.) continue;
    sd = sqrt(d);
    s = (dab+sd)/na;
    t = (dab-sd)/na;
    if ((s>= 0.&&s<=1.) || (t>=0.&&t<=1.)) {
      return true;
    }
  }
  return false;
}

int main() {
  while (1) {
    cin >> v[0].real() >> v[0].imag();
    if (!v[0].real() && !v[0].imag()) break;
    for (int i=1; i<4; i++) {
      cin >> v[i].real() >> v[i].imag();
    }
    cin >> r;
    if (tri_in_check()) {
      cout << "b" << endl;
    } else {
      if (intersect()) {
        cout << "c" << endl;
      } else if (center_in_check()) {
        cout << "a" << endl;
      } else {
        cout << "d" << endl;
      }
    }
  }
  return 0;
}