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]);
}

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 <= EPS) {
      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() == -1) {
      cout << "b" << endl;
    } else {
      if (intersect()) {
        cout << "c" << endl;
      } else if (center_in_check()) {
        cout << "a" << endl;
      } else {
        cout << "d" << endl;
      }
    }
  }
  return 0;
}

C++: #include <iostream>
#include <sstream>
#include <cstdio>
#include <cstdlib>
#include <cmath>
#include <ctime>
#include <cstring>
#include <climits>
#include <algorithm>
#include <map>
#include <set>
#include <stack>
#include <vector>
#include <queue>
#include <string>
#include <complex>
using namespace std;

#define rep(i,n) for(int i=0; i<n; i++)
#define repa(i,s,e) for(int i=s; i<=e; i++)
#define repd(i,s,e) for(int i=s; i>=e; i--)

typedef long long ll;
typedef vector<int> vi;
typedef pair<int, int> pii;

const double EPS = 1.0e-10;

class P {
public:
	double x, y;
	P() {}
	P(double x_, double y_) : x(x_), y(y_) {}
	P(const P& p) : x(p.x), y(p.y) {}
	P operator+(const P& p) const { return P(x+p.x, y+p.y); } 
	P operator-(const P& p) const { return P(x-p.x, y-p.y); }
	P operator-() const { return P(-x, -y); }
	P operator*(double s) const { return P(x*s, y*s); }
	P operator/(double s) const { return P(x/s, y/s); }
	P& operator=(const P& p) { x=p.x; y=p.y; return (*this); }
	double dot(const P& p) const { return x*p.x + y*p.y; }
	double det(const P& p) const { return x*p.y - y*p.x; }
	double norm() const { return sqrt(x*x + y*y); }
	double norm2() const { return x*x + y*y; }
};

double tri(P a, P b, P c) { return (b - a).det(c - a); }
double sign(double r) { return r < -EPS ? -1.0 : r > EPS ? 1.0 : 0.0; }

class L {
public:
	P s, t;
	L() {}
	L(P s_, P t_) : s(s_), t(t_) {}
	L(const L& l) : s(l.s), t(l.t) {}
	L& operator=(const L& l) { s=l.s; t=l.t; return (*this); }
	double length() { return (s - t).norm(); }
	P vec() const { return t - s; }
	double dLP(const P& p) const { return abs(tri(s, t, p)) / vec().norm(); }
	double dSP(const P& p) const {
		if(sign(vec().dot(p - s)) <= 0.0) return (p - s).norm();
		if(sign(vec().dot(p - t)) <= 0.0) return (p - t).norm();
		return dLP(p);
	}
};

class C {
public:
	P p;
	double r;
	C() {}
	C(P p_, double r_) : p(p_), r(r_) {}
	C(const C& c) : p(c.p), r(c.r) {}
	C& operator=(const C& c) { p=c.p; r=c.r; return (*this); }
};


// code for problem
int x, y, r;
vector<P> T;
C cc;

bool isPinC() {
	P e0 = T[0] - cc.p;
	P e1 = T[1] - cc.p;
	P e2 = T[2] - cc.p;
	double d0 = e0.det(e1);
	double d1 = e1.det(e2);
	double d2 = e2.det(e0);
	if(sign(d0) == sign(d1) && sign(d0) == sign(d2)) return true;
	return false;
}

void solve() {
	vector<double> d(3);
	d[0] = (T[0] - cc.p).norm();
	d[1] = (T[1] - cc.p).norm();
	d[2] = (T[2] - cc.p).norm();
	if(sign(d[0] - cc.r) <= 0.0 && sign(d[1] - cc.r) <= 0.0 && sign(d[2] - cc.r) <= 0.0) {
		printf("b\n");
		return;
	}

	bool in = isPinC();
	L l0(T[0], T[1]);
	L l1(T[1], T[2]);
	L l2(T[2], T[0]);
	d[0] = l0.dSP(cc.p);
	d[1] = l1.dSP(cc.p);
	d[2] = l2.dSP(cc.p);
	if(in && sign(d[0] - cc.r) >= 0.0 && sign(d[1] - cc.r) >= 0.0 && sign(d[2] - cc.r) >= 0.0) {
		printf("a\n");
		return;
	}

	sort(d.begin(), d.end());
	if(!in && d[0] > cc.r) {
		printf("d\n");
		return;
	}

	printf("c\n");
}

int main() {
	T = vector<P>(3);
	for(;;) {
		scanf("%d%d", &x, &y);
		if(x == 0 && y == 0) break;
		T[0] = P(x, y);

		scanf("%d%d", &x, &y);
		T[1] = P(x, y);

		scanf("%d%d", &x, &y);
		T[2] = P(x, y);

		scanf("%d%d%d", &x, &y, &r);
		cc = C(P(x, y), r);

		solve();
	}
}