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<algorithm>
#include<iostream>
#include<string>
#include<cstdlib>
#include<cstring>
#include<cmath>
#define dis(p,q) ((p)*(p)+(q)*(q))
using namespace std;
int lc(double bx,double by,double ex,double ey,double cx,double cy,double cr){
  int p,q;
  if(0){
  }else if(dis(cx-bx,cy-by)>cr*cr){
    p=1;
  }else if(dis(cx-bx,cy-by)<cr*cr){
    p=0;
  }else{
    p=2;
  }
  if(0){
  }else if(dis(cx-ex,cy-ey)>cr*cr){
    q=1;
  }else if(dis(cx-ex,cy-ey)<cr*cr){
    q=0;
  }else{
    q=2;
  }
  if(p==0&&q==0)
    return 0;
  if(p!=1||q!=1)
    return 1;
  double vx,vy,px,py,qx,qy,rx,ry,ds;
  ds=sqrt(dis(bx-ex,by-ey));
  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){
    ds=sqrt(cr*cr-ds*ds);
    qx=px+vx*ds;
    qy=py+vy*ds;
    rx=px-vx*ds;
    ry=py-vy*ds;
    if(bx==ex)
      return min(by,ey)<=min(qy,ry)&&max(qy,ry)<=max(by,ey);
    else
      return min(bx,ex)<=min(qx,rx)&&max(qx,rx)<=max(bx,ex);
  }else if(cr==ds){
    if(bx==ex)
      return min(by,ey)<=py&&py<=max(by,ey);
    else
      return min(bx,ex)<=px&&px<=max(bx,ex);
  }
  return 0;
}
int pt(double px,double py,double qx,double qy,double rx,double ry,double sx,double sy){
  if(((rx-qx)*(py-qy)-(px-qx)*(ry-qy)<=0&&(sx-rx)*(py-ry)-(px-rx)*(sy-ry)<=0&&(qx-sx)*(py-sy)-(px-sx)*(qy-sy)<=0)||
     ((rx-qx)*(py-qy)-(px-qx)*(ry-qy)>=0&&(sx-rx)*(py-ry)-(px-rx)*(sy-ry)>=0&&(qx-sx)*(py-sy)-(px-sx)*(qy-sy)>=0))
    return 1;
  return 0;
}
int main(){
  int i;
  int x[4],y[4],r;
  while(cin>>x[0]>>y[0]&&(x[0]||y[0])){
    for(i=1;i<4;i++)
      cin>>x[i]>>y[i];
    cin>>r;
    for(i=0;i<3;i++){
      if(lc(x[i],y[i],x[(i+1)%3],y[(i+1)%3],x[3],y[3],r))
	break;
    }
    if(i==3){
      if(dis(x[0]-x[3],y[0]-y[3])<r*r){
	cout<<"b"<<endl;
      }else{
	if(pt(x[3],y[3],x[0],y[0],x[1],y[1],x[2],y[2]))
	  cout<<"a"<<endl;
	else
	  cout<<"d"<<endl;
      }
    }else{
      cout<<"c"<<endl;
    }
  }
  return 0;
}