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 <map>
#include <cmath>

using namespace std;

const double EPS = 1e-10;

pair<int, int> a;
pair<int, int> b;
pair<int, int> c;
pair<int, int> o;
int r;

int dot(pair<int, int> p1, pair<int, int> p2){
  return p1.first * p2.first + p1.second * p2.second;
}

int cross(pair<int, int> p1, pair<int, int> p2){
  return p1.first * p2.second - p1.second * p2.first;
}

double length(pair<int, int> p){
  return sqrt(p.first * p.first + p.second * p.second);
}

bool is_inside(){
  if(cross(make_pair(b.first-a.first, b.second-a.second), 
	   make_pair(o.first-a.first, o.second-a.second)) < 0) return false;
  if(cross(make_pair(c.first-b.first, c.second-b.second), 
	   make_pair(o.first-b.first, o.second-b.second)) < 0) return false;
  if(cross(make_pair(a.first-c.first, a.second-c.second), 
	   make_pair(o.first-c.first, o.second-c.second)) < 0) return false;
  return true;
}

double dist(pair<int, int> p1, pair<int, int> p2, pair<int, int> p3){
  pair<int, int> p4 = make_pair(p2.first - p1.first, p2.second - p1.second);
  pair<int, int> p5 = make_pair(p3.first - p1.first, p3.second - p1.second);
  double tmp = length(p4);
  return abs((double)cross(p4, p5) / tmp);
}

double dist2(pair<int, int> p1, pair<int, int> p2, pair<int, int> p3){
  pair<int, int> p7 = make_pair(p2.first - p1.first, p2.second - p1.second);
  pair<int, int> p4 = make_pair(p1.first - p2.first, p1.second - p2.second);
  pair<int, int> p5 = make_pair(p3.first - p2.first, p3.second - p2.second);
  pair<int, int> p6 = make_pair(p3.first - p1.first, p3.second - p1.second);
  if(dot(p7, p6) < 0) return length(p6)*length(p6);
  if(dot(p4, p5) < 0) return length(p5)*length(p5);
  return dist(p1, p2, p3);
}

main(){
  while(cin >> a.first >> a.second && a.first && a.second){
    cin >> b.first >> b.second;
    cin >> c.first >> c.second;
    cin >> o.first >> o.second;
    cin >> r;
    if(length(make_pair(a.first-o.first, a.second-o.second)) < (double)r+EPS && 
       length(make_pair(b.first-o.first, b.second-o.second)) < (double)r+EPS &&
       length(make_pair(c.first-o.first, c.second-o.second)) < (double)r+EPS){
      cout << "b" << endl;
    }
    else if(is_inside()){
      if(dist(b, c, o) > (double)r-EPS && 
	 dist(a, c, o) > (double)r-EPS && 
	 dist(a, b, o) > (double)r-EPS){
	cout << "a" << endl;
      }else{
	cout << "c" << endl;
      }
    }else{
      if(dist2(b, c, o) < (double)r+EPS ||
	 dist2(a, c, o) < (double)r+EPS ||
	 dist2(a, b, o) < (double)r+EPS){
	cout << "c" << endl;
      }else{
	cout << "d" << endl;
      }
    }
  }
  return 0;
}

C++: #include<bits/stdc++.h>
using namespace std;
#define rep(i,n) for (int i=0; i < (n); i++)
typedef double D;      // ??§?¨?????????????double???long double?????????
typedef complex<D> P;  // Point
typedef pair<P, P> L;  // Line
typedef vector<P> VP;
const D EPS = 1e-9;    // ?¨±??????????????????????????£????????????
#define X real()
#define Y imag()
#define LE(n,m) ((n) < (m) + EPS)
#define GE(n,m) ((n) + EPS > (m))
#define EQ(n,m) (abs((n)-(m)) < EPS)
D dot(P a, P b) {return (conj(a)*b).X; }
D cross(P a, P b) {  return (conj(a)*b).Y;}
int ccw(P a, P b, P c) {
  b -= a;  c -= a;
  if (cross(b,c) >  EPS) return +1;  // counter clockwise
  if (cross(b,c) < -EPS) return -1;  // clockwise
  if (dot(b,c)   < -EPS) return +2;  // c--a--b on line
  if (norm(b) < norm(c)) return -2;  // a--b--c on line or a==b
  return 0;                          // a--c--b on line or a==c or b==c
}
bool isecSP(P a1, P a2, P b) {
  return !ccw(a1, a2, b);
}
P proj(P a1, P a2, P p) {
  return a1 + dot(a2-a1, p-a1)/norm(a2-a1) * (a2-a1);
}
D distSP(P a1, P a2, P p) {
  P r = proj(a1, a2, p);
  if (isecSP(a1, a2, r)) return abs(r-p);
  return min(abs(a1-p), abs(a2-p));
}

D dis(P a,P b){
	return sqrt((a.X-b.X)*(a.X-b.X)+(a.Y-b.Y)*(a.Y-b.Y));
}
double x1,x2,x3,y4,y2,y3,x,y,r;
main(){
	while(cin>>x1>>y4,x1||y4){
		cin>>x2>>y2>>x3>>y3>>x>>y>>r;
		if(dis(P(x1,y4),P(x,y))<=r&&dis(P(x2,y2),P(x,y))<=r&&dis(P(x3,y3),P(x,y))<=r)cout<<'b'<<endl;
		else if(distSP(P(x1,y4),P(x2,y2),P(x,y))<=r||distSP(P(x3,y3),P(x2,y2),P(x,y))<=r||distSP(P(x1,y4),P(x3,y3),P(x,y))<=r)cout<<'c'<<endl;
		else if(ccw(P(x1,y4),P(x2,y2),P(x,y))==ccw(P(x2,y2),P(x3,y3),P(x,y))==ccw(P(x3,y3),P(x1,y4),P(x,y)))cout<<'a'<<endl;
		else cout<<'d'<<endl;
	}
}