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-8
typedef long long int lli;
typedef complex<lli> C;

C v[4];
lli r;

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

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

int tri_in_check() {
  int c[3];
  C a;
  lli ina;
  lli ir2 = r*r;
  for (int i=0; i<3; i++) {
    a = v[i] - v[3];
    ina = norm(a);
    if (ina < ir2) {
      c[i] = -1;
    } else if (ina > ir2){
      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;
  lli cab;
  bool f[3];
  for (int i=0; i<3; i++) {
    a = v[(i+1)%3] - v[i];
    b = v[3] - v[(i+1)%3];
    cab = zcross(a,b);
    f[i] = (cab > 0);
    if (cab*cab < norm(a)*r*r) {
      return false;
    }
  }
  return (f[0]==f[1] && f[0]==f[2]);
}

int intersect() {
  C a, b;
  double sd, s, t;
  long long int ina, inb, idab, id;
  bool contact = false;
  for (int i=0; i<3; i++) {
    a = v[(i+1)%3] - v[i];
    b = v[3] - v[i];
    ina = norm(a);
    inb = norm(b);
    idab = dot(a,b);
    id = idab*idab - ina*(inb-r*r);
    if (id < 0) {
      continue;
    }
    sd = sqrt((double)id);
    s = (idab+sd)/ina;
    t = (idab-sd)/ina;
    if ((s>EPS&&s<1.-EPS) || (t>EPS&&t<1.-EPS)) {
      if (!id) {
        contact = true;
      } else {
        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 <sstream>
#include <string>
#include <algorithm>
#include <vector>
#include <map>
#include <set>
#include <bitset>
#include <queue>
#include <stack>
#include <utility>
#include <complex>
#include <cstring>
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <cctype>
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 vector<int> vi;
typedef long long ll;
typedef pair<ll,ll> pll;

typedef complex<ll> P;

int x, y, r;
P tri[3];
pair<P,ll> circle;

ll cross(P e1, P e2) {
	return e1.real() * e2.imag() - e1.imag() * e2.real();
}

bool inc() {
	P e0 = tri[0] - circle.first;
	P e1 = tri[1] - circle.first;
	P e2 = tri[2] - circle.first;
	ll d0 = cross(e0, e1);
	ll d1 = cross(e1, e2);
	ll d2 = cross(e2, e0);
	if(d0 * d1 > 0 && d1 * d2 > 0 && d2 * d0 > 0) return true;
	return false;
}

ll norm2(P p) {
	return p.real() * p.real() + p.imag() * p.imag();
}

void dist2(P p1, P p2, P p, ll& nm, ll& de) {
	ll a = p2.imag() - p1.imag();
	ll b = -(p2.real() - p1.real());
	ll c = - a * p1.real() - b * p1.imag();
	nm = a * p.real() + b * p.imag() + c;
	nm = nm * nm;
	de = a * a + b * b;
	if(de == 0) {
		nm = norm2(p1 - p);
		de = 1;
	}
}


void solve() {
	ll r2 = circle.second * circle.second;
	vector<pll> d(6);
	d[3].first = norm2(tri[0] - circle.first); d[3].second = 1LL;
	d[4].first = norm2(tri[1] - circle.first); d[4].second = 1LL;
	d[5].first = norm2(tri[2] - circle.first); d[5].second = 1LL;
	if(d[3].first <= r2 && d[4].first <= r2 && d[5].first <= r2) {
		cout << "b" << endl;
		return;
	}

	bool iscin = inc();
	dist2(tri[0], tri[1], circle.first, d[0].first, d[0].second);
	dist2(tri[1], tri[2], circle.first, d[1].first, d[1].second);
	dist2(tri[2], tri[0], circle.first, d[2].first, d[2].second);
	if(iscin && (d[0].first >= r2 * d[0].second) &&
		        (d[1].first >= r2 * d[1].second) &&
		        (d[2].first >= r2 * d[2].second)) {
		cout << "a" << endl;
		return;
	}

	bool inter = false;
	rep(i,6) {
		if(d[i].first <= r2 * d[i].second) {
			inter = true;
			break;
		}
	}

	if(inter) {
		cout << "c" << endl;
		return;
	}

	cout << "d" << endl;
}

int main() {
	while( 1 ) {
		scanf("%d%d", &x, &y);
		if(x == 0 && y == 0) break;
		tri[0] = P(x, y);

		scanf("%d%d", &x, &y);
		tri[1] = P(x, y);
		
		scanf("%d%d", &x, &y);
		tri[2] = P(x, y);
		
		scanf("%d%d", &x, &y);
		scanf("%d", &r);
		circle.first = P(x, y);
		circle.second = r;

		solve();
	}
}