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 <sstream>
#include <string>
#include <algorithm>
#include <vector>
#include <stack>
#include <queue>
#include <complex>
#include <set>
#include <map>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <cmath>
#include <cassert>

using namespace std;

#define FOR(i,k,n) for(int i=(k); i<(int)(n); ++i)
#define REP(i,n) FOR(i,0,n)
#define FORIT(i,c) for(__typeof((c).begin())i=(c).begin();i!=(c).end();++i)

template<class T> void debug(T begin, T end){ for(T i = begin; i != end; ++i) cerr<<*i<<" "; cerr<<endl; }
inline bool valid(int x, int y, int W, int H){ return (x >= 0 && y >= 0 && x < W && y < H); }

typedef long long ll;
const int INF = 100000000;
const double EPS = 1e-8;
const int MOD = 1000000007;
int dx[8] = {1, 0, -1, 0, 1, -1, -1, 1};
int dy[8] = {0, 1, 0, -1, 1, 1, -1, -1};

typedef complex<double> P;
typedef pair<P, P> L;
#define X real()
#define Y imag()

double dot(P& a, P& b){
  return a.X * b.X + a.Y * b.Y;
}
double cross(P& a, P& b){
  return a.X * b.Y - a.Y * b.X;
}

int sign(double x){
  if(x > EPS) return 1;
  if(x < -EPS) return -1;
  return 0;
}

//a -> b, a -> c
int ccw(P a, P b, P c){
  b -= a; c -= a;
  if(sign(cross(b, c)) > 0) return 1; //反時計
  if(sign(cross(b, c)) < 0) return -1; //時計
  if(sign(dot(b, c)) < 0) return +2; //c-a-b
  if(norm(b) < norm(c)) return -2; //a-b-c
  return 0; //a-c-b
}

double min_d(P a, P b, P c){
  double d1 = abs(c - a);
  double d2 = abs(c - b);
  b -= a; c -= a;
  double k = dot(b, c) / norm(b);
  if(0 < k && k < 1) return abs(c - k * b);
  else return min(d1, d2);
}

string solve(P p[3], P c, double r){
  if(ccw(p[0], p[1], p[2]) == -1) swap(p[1], p[2]);
  double maxdist = 0, mindist = 1e18;
  int ccw_v[3];
  REP(i, 3){
    maxdist = max(maxdist, abs(p[i] -  c));
    mindist = min(mindist, min_d(p[i], p[(i + 1) % 3], c));
    ccw_v[i] = ccw(p[i], p[(i + 1) % 3], c);
  }
  if(maxdist < r) return "b";
  if(ccw_v[0] == ccw_v[1] && ccw_v[1] == ccw_v[2] && ccw_v[0] == 1){
    if(mindist >= r + EPS) return "a";
    else return "c";
  }else{
    if(mindist > r) return "d";
    else return "c";
  }
}

int main(){
  P p[3];
  P c; double r;
  while(cin>>p[0].X>>p[0].Y>>p[1].X>>p[1].Y>>p[2].X>>p[2].Y>>c.X>>c.Y>>r){
    cout<<solve(p, c, r)<<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;
	}
}