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++: #define _USE_MATH_DEFINES
#define INF 0x3f3f3f3f
#include <cstdio>
#include <iostream>
#include <sstream>
#include <cmath>
#include <cstdlib>
#include <algorithm>
#include <queue>
#include <stack>
#include <limits>
#include <map>
#include <string>
#include <cstring>
#include <set>
#include <deque>
#include <bitset>
#include <list>
#include <cctype>
#include <utility>
#include <complex>
#include <assert.h>
 
using namespace std;
 
typedef long long ll;
typedef pair <int,int> P;
typedef pair <int,P > PP;
 

static const int tx[] = {0,1,0,-1};
static const int ty[] = {-1,0,1,0};
 
static const double EPS = 1e-10;

typedef complex<double> Point;

namespace std {
  bool operator < (const Point& a, const Point& b) {
    return real(a) != real(b) ? real(a) < real(b) : imag(a) < imag(b);
  }
}

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

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

int ccw(Point a, Point b, Point c) {
  b -= a; c -= a;
  if (cross(b, c) > 0)   return +1;       // counter clockwise
  if (cross(b, c) < 0)   return -1;       // clockwise
  if (dot(b, c) < 0)     return +2;       // c--a--b on line
  if (norm(b) < norm(c)) return -2;       // a--b--c on line
  return 0;
}

bool is_inner(Point a, Point b, Point c) {
  b -= a; c -= a;
  return (dot(b,c) <= 0);
}

bool intersectSS(const Line &s, const Line &t) {
  return ccw(s[0],s[1],t[0])*ccw(s[0],s[1],t[1]) <= 0 &&
    ccw(t[0],t[1],s[0])*ccw(t[0],t[1],s[1]) <= 0;
}

Point crosspoint(const Line &l, const Line &m) {
  double A = cross(l[1] - l[0], m[1] - m[0]);
  double B = cross(l[1] - l[0], l[1] - m[0]);
  if (abs(A) < EPS && abs(B) < EPS) return m[0]; // same line
  if (abs(A) < EPS) assert(false); // !!!PRECONDITION NOT SATISFIED!!!
  return m[0] + B / A * (m[1] - m[0]);
}

Point projection(const Line &l, const Point &p) {
  double t = dot(p-l[0], l[0]-l[1]) / norm(l[0]-l[1]);
  return l[0] + t*(l[0]-l[1]);
}

bool is_projection_on_line(const Line &l, const Point &p) {
  Point pj = projection(l,p);
  return is_inner(pj,l[0],l[1]);
}

double distanceLP(const Line &l, const Point &p) {
  return abs(p - projection(l, p));
}

bool is_equal(const Point &l,const Point &m){
  return ((abs(real(l) - real(m)) < EPS) && (abs(imag(l) - imag(m) < EPS)));
}

bool cmp_x(const Point& p, const Point& q){
  if(p.real() != q.real()) return p.real() < q.real();
  return p.imag() < q.imag();
}

vector<Point> compute_convex_hull(vector<Point> ps){
  sort(ps.begin(), ps.end(), cmp_x);
  int k = 0;
  vector<Point> qs(ps.size() * 2);
  for(int i=0; i < ps.size(); i++){
    while(k > 1 && cross(qs[k - 1] - qs[k - 2],ps[i] - qs[k - 1]) <= 0) k--;
    qs[k++] = ps[i];
  }
  for(int i = ps.size() - 2,t = k; i >= 0; i--){
    while(k > t && cross(qs[k - 1] - qs[k - 2],ps[i] - qs[k - 1]) <= 0) k--;
    qs[k++] = ps[i];
  }
  qs.resize(k-1);
  return qs;
}

bool is_convex_hull(vector<Point>& ps,const Point& c){
  for(int i=0;i<ps.size();i++){
    if(is_equal(ps[i],c)) return true;
  }
  return false;
}

int main(){
  double x[4];
  double y[4];
  while(~scanf("%lf %lf",&x[0],&y[0])){
    if(is_equal(Point(x[0],y[0]),Point(0.0,0.0))) break;

    for(int i=1;i<4;i++){
      scanf("%lf %lf",&x[i],&y[i]);
    }
    double r;
    scanf("%lf",&r);

    vector<Line> lines;
    vector<Point> points;

    for(int i=0;i<4;i++){
      points.push_back(Point(x[i],y[i]));
    }

    for(int i=0;i<3;i++){
      for(int j=i+1;j<3;j++){
        lines.push_back(Line(Point(x[i],y[i]),Point(x[j],y[j])));
      }
    }

    vector<Point> convex_hull = compute_convex_hull(points);

    if(!is_convex_hull(convex_hull,points[3])){
      double dist2line = numeric_limits<double>::max();
      for(int i=0;i<3;i++){
        dist2line = min(distanceLP(lines[i],Point(x[3],y[3])),dist2line);
      }
      
      if(r <= dist2line + EPS){
        //a
        printf("a\n");
      }
      else{
        double dist2point = numeric_limits<double>::min();
        for(int i=0;i<3;i++){
          dist2point = max(sqrt(dot(points[i]-points[3],points[i]-points[3])),dist2point);
        }
        if(r >= dist2point - EPS){
          //b
          printf("b\n");
        }
        else{
          //c
          printf("c\n");
        }
      }
    }
    else {
      double dist2LP = numeric_limits<double>::max();
      for(int i=0;i<3;i++){
        if(is_projection_on_line(lines[i],Point(x[3],y[3]))){
          dist2LP = min(distanceLP(lines[i],Point(x[3],y[3])),dist2LP);
        }
      }
      for(int i=0;i<3;i++){
        dist2LP = min(sqrt(dot(points[i]-points[3],points[i]-points[3])),dist2LP);
      }
      
      if(dist2LP > r){
        //d
        printf("d\n");
      }
      else{
        double dist2point = numeric_limits<double>::min();
        for(int i=0;i<3;i++){
          dist2point = max(sqrt(dot(points[i]-points[3],points[i]-points[3])),dist2point);
        }
        
        if(r >= dist2point - EPS){
          //b
          printf("b\n");
        }
        else{
          //c
          printf("c\n");
        }
      }
    }
  }
}