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;
      continue;
    }
    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;
	continue;
      }
    }
    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;
      continue;
    }
    cout << "d" << endl;
  }
  return 0;
}

C++: 
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <cmath>
#include <ctime>
#include <cassert>
#include <iostream>
#include <cctype>
#include <sstream>
#include <string>
#include <list>
#include <vector>
#include <queue>
#include <set>
#include <stack>
#include <map>
#include <utility>
#include <numeric>
#include <algorithm>
#include <iterator>
#include <bitset>
#include <complex>
#include <fstream>
using namespace std;
typedef long long ll;
const double EPS = 1e-9;
typedef vector<int> vint;
typedef pair<int, int> pint;
#define rep(i, n) REP(i, 0, n)
#define ALL(v) v.begin(), v.end()
#define MSG(a) cout << #a << " " << a << endl;
#define REP(i, x, n) for(int i = x; i < n; i++)
template<class T> T RoundOff(T a){ return int(a+.5-(a<0)); }
template<class T, class C> void chmax(T& a, C b){ if(a < b) a = b; }
template<class T, class C> void chmin(T& a, C b){ if(b < a) a = b; }
template<class T, class C> pair<T, C> mp(T a, C b){ return make_pair(a, b); }

typedef complex<double> P;

// OÏ
double cross(P a, P b)
{	return a.real() * b.imag() - b.real() * a.imag();	}

// àÏ
double dot(P a, P b)
{	return a.real() * b.real() + a.imag() * b.imag();	}

// _ÆüªÌ£
double dist(P a, P b, P c)
{
	if(dot(b - a, c - a) < EPS) return abs(c - a);
	if(dot(a - b, c - b) < EPS) return abs(c - b);
	
	return abs(cross(b - a, c - a)) / abs(b - a);
}

bool LineCrossCircle(P a, P b, P c, double r)
{
	return dist(a, b, c) <= r && (r < abs(a - c) || r < abs(b - c));
}

bool CircleInTriangle(P p[3], P c, double r)
{
	rep(i, 3) if(dist(p[i], p[(i + 1) % 3], c) <= r) return false;
	
	double S = abs(cross(p[0] - p[2], p[1] - p[2]));
	
	rep(i, 3) S -= abs(cross(p[(i + 1) % 3] - p[i], c - p[i]));
	
	return abs(S) < EPS;
}

bool TriangleInCircle(P p[3], P c, double r)
{
	rep(i, 3) if(r < abs(p[i] - c)) return false;
	
	return true;
}

bool TriangleCrossCircle(P p[3], P c, double r)
{
	rep(i, 3) if(LineCrossCircle(p[i], p[(i + 1) % 3], c, r)) return true;	
	
	return false;
}

int main()
{
    P p[3], c;
	double r;
	
	while(cin >> p[0].real() >> p[0].imag() && (p[0].real() || p[0].imag()))
	{
		rep(i, 2) cin >> p[i + 1].real() >> p[i + 1].imag();
		
		cin >> c.real() >> c.imag() >> r;
		
		if(CircleInTriangle(p, c, r))
		{
			cout << "a" << endl;
		}
		else if(TriangleInCircle(p, c, r))
		{
			cout << "b" << endl;
		}
		else if(TriangleCrossCircle(p, c, r))
		{
			cout << "c" << endl;
		}
		else
		{
			cout << "d" << endl;
		}
	}
}