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<cmath>
#include<algorithm>
#include<vector>
using namespace std;
struct Point { long double px, py; };
struct Line { Point p1, p2; };
Point Minus(const Point& a, const Point& b) { return Point{ a.px - b.px,a.py - b.py }; }
long double norm(const Point& a) { return a.px * a.px + a.py * a.py; }
long double abs(const Point& a) { return sqrtl(norm(a)); }
long double dot(const Point& a, const Point& b) { return a.px * b.px + a.py * b.py; }
long double crs(const Point& a, const Point& b) { return a.px * b.py - a.py * b.px; }
long double dst(const Point& a, const Point& b) { return abs(Minus(b , a)); }
long double dst(const Line& a, const Point& b) {
	return dot(Minus(a.p2, a.p1), Minus(b, a.p1)) < 0.0 ? abs(Minus(b, a.p1)) : (dot(Minus(a.p1, a.p2), Minus(b, a.p2)) < 0.0 ? abs(Minus(b, a.p2)) : abs(crs(Minus(a.p2, a.p1), Minus(b, a.p1)) / abs(Minus(a.p2, a.p1))));
}
int contain(vector<Point> v, Point p) {
	bool in = false;
	for (int i = 0; i < v.size(); ++i) {
		Point a = Minus(v[i], p), b = Minus(v[(i + 1) % v.size()], p);
		if (a.py > b.py) swap(a, b);
		if (a.py <= 1e-7 && 1e-7 < b.py)
			if (crs(a, b) < -1e-7) in = !in;
		if (fabs(crs(a, b)) <= 1e-7 && dot(a, b) <= 1e-7) return 1;
	}
	return in ? 2 : 0;
}
int main() {
	while (true) {
		Point r1, r2, r3, r4; long double r;
		cin >> r1.px >> r1.py; if (fabs(r1.px) <= 1e-7 && fabs(r1.py) <= 1e-7)break;
		cin >> r2.px >> r2.py >> r3.px >> r3.py >> r4.px >> r4.py >> r;
		Line l1 = Line{ r1,r2 }, l2 = { r2,r3 }, l3 = { r3,r1 };
		long double c1 = min(dst(l1, r4), min(dst(l2, r4), dst(l3, r4)));
		long double c2 = max(dst(r1, r4), max(dst(r2, r4), dst(r3, r4)));
		long double c3 = min(dst(r1, r4), min(dst(r2, r4), dst(r3, r4)));
		//cout << dst(l1, r4) << ' ' << dst(l2, r4) << ' ' << dst(l3, r4) << endl;
		int c4 = contain({ r1,r2,r3 }, r4);
		if (c1 >= r - 1e-7 && c4 != 0) { cout << "a" << endl; }
		else if (c2 <= r + 1e-7) { cout << "b" << endl; }
		else if (c3 <= r + 1e-7) { cout << "c" << endl; }
		else { cout << "d" << endl; }
	}
	return 0;
}

C++: #include<algorithm>
#include<iostream>
#include<string>
#include<cstdio>
#include<cstdlib>
#include<cstring>
#include<cmath>
using namespace std;
void f(double& v,double& w,double& x,double& y,int& n,int bx,int by,int ex,int ey,int cx,int cy,int cr){
  double vx,vy,px,py,ds;
  ds=sqrt((ex-bx)*(ex-bx)+(ey-by)*(ey-by));
  vx=(ex-bx)/ds;
  vy=(ey-by)/ds;
  ds=((ey-by)*cx+(bx-ex)*cy+(by-ey)*bx+(ex-bx)*by)/((ey-by)*vy-(bx-ex)*vx);
  px=cx-vy*ds;
  py=cy+vx*ds;
  if(0){
  }else if(cr>ds){
    n=2;
    ds=sqrt(cr*cr-ds*ds);
    v=px+vx*ds;
    w=py+vy*ds;
    x=px-vx*ds;
    y=py-vy*ds;
  }else if(cr<ds){
    n=0;
  }else{
    n=1;
    v=px;
    w=py;
  }
  return;
}
int g(int bx,int by,int ex,int ey,int cx,int cy,int cr){
  int n;
  double v,w,x,y;
  f(v,w,x,y,n,bx,by,ex,ey,cx,cy,cr);
  if(0){
  }else if(n==0){
    return 0;
  }else if(n==1){
    if(bx==ex)
      return (min(by,ey)<w&&w<max(by,ey))*2;
    else
      return (min(bx,ex)<v&&v<max(bx,ex))*2;
  }else{
    if(bx==ex)
      return min(by,ey)<min(w,y)&&max(w,y)<max(by,ey);
    else
      return min(bx,ex)<min(v,x)&&min(v,x)<max(bx,ex);
  }
}
int range(double p,double q,double r){
  return (q<r?q:r)<p&&p<(q<r?r:q);
}
int rangea(double p,double q,double r){
  if(q*r<0&&fabs(q)+fabs(r)>M_PI){
    p+=2*M_PI*(p<0);
    q+=2*M_PI*(q<0);
    r+=2*M_PI*(r<0);
  }
  return range(p,q,r);
}
int pit(int ax,int ay,int bx,int by,int cx,int cy,int px,int py){
  if(rangea(atan2(ay-py,ax-px),atan2(ay-by,ax-bx),atan2(ay-cy,ax-cx))){
    if(rangea(atan2(by-py,bx-px),atan2(by-ay,bx-ax),atan2(by-cy,bx-cx))){
      return 1;
    }
  }
  return 0;
}
int main(){
  int i;
  int ac[3],ax[3],ay[3],cx,cy,cr;
  while(cin>>ax[0]>>ay[0]&&ax[0]&&ay[0]){
    for(i=1;i<3;i++)
      cin>>ax[i]>>ay[i];
    cin>>cx>>cy>>cr;
    for(i=0;i<3;i++)
      ac[i]=(ax[i]-cx)*(ax[i]-cx)+(ay[i]-cy)*(ay[i]-cy)>cr*cr;
    if(0){
    }else if(ac[0]&ac[1]&ac[2]){
      for(i=0;i<3;i++)
	ac[i]=g(ax[i],ay[i],ax[(i+1)%3],ay[(i+1)%3],cx,cy,cr);
      if(0){
      }else if(ac[0]==0&&ac[1]==0&&ac[2]==0){
	if(pit(ax[0],ay[0],ax[1],ay[1],ax[2],ay[2],cx,cy))
	  cout<<"a"<<endl;
	else
	  cout<<"d"<<endl;
      }else if(ac[0]==1||ac[1]==1||ac[2]==1){
	cout<<"c"<<endl;
      }else{
	if(pit(ax[0],ay[0],ax[1],ay[1],ax[2],ay[2],cx,cy))
	  cout<<"a"<<endl;
	else
	  cout<<"c"<<endl;
      }
    }else if(ac[0]|ac[1]|ac[2]){
      cout<<"c"<<endl;
    }else{
      cout<<"b"<<endl;
    }
  }
  return 0;
}