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>
using namespace std;

int x[3], y[3], xc, yc, r, d[4];
const double INF = 10000000;

bool in() {
	bool f[3];

	for (int i = 0; i < 3; i++) {
		int j = (i + 1) % 3, k = (i + 2) % 3;
		f[i] = ((x[i] - x[j]) * (yc - y[i]) - (y[i] - y[j]) * (xc - x[i])) * ((x[i] - x[j]) * (y[k] - y[i]) - (y[i] - y[j]) * (x[k] - x[i])) >= 0;
	}

	return f[0] && f[1] && f[2];
}

double minds() {
	double g[3];
	for (int i = 0; i < 3; i++) {
		int j = (i + 1) % 3;

		if ((x[j] - x[i]) * (xc - x[i]) + (y[j] - y[i]) * (yc - y[i]) >= 0
			&& (x[i] - x[j]) * (xc - x[j]) + (y[i] - y[j]) * (yc - y[j]) >= 0)
			g[i] = pow((y[j] - y[i]) * xc - (x[j] - x[i]) * yc - x[i] * y[j] + y[i] * x[j], 2) / (pow(y[j] - y[i], 2) + pow(x[j] - x[i], 2));
		else
			g[i] = INF;
	}
		return min({ g[0], g[1], g[2] });
}

int main()
{
	cin >> x[0] >> y[0];

	while (x[0] != 0 || y[0] != 0) {
		for (int i = 1; i < 3; i++)
			cin >> x[i] >> y[i];

		cin >> xc >> yc >> r;

		for (int i = 0; i < 3; i++)
			d[i] = (int)(pow(x[i] - xc, 2) + pow(y[i] - yc, 2));
		
		int rs = r * r;

		if (rs >= d[0] && rs >= d[1] && rs >= d[2])
			cout << "b" << endl;
		else if (in() && minds() >= rs)
			cout << "a" << endl;
		else if (!in() && rs < d[0] && rs < d[1] && rs < d[2] && minds() > rs)
			cout << "d" << endl;
		else
			cout << "c" << endl;

		cin >> x[0] >> y[0];
	}

	return 0;
}

C++: #include <iostream>
#include <stdio.h>
#include <math.h>
#define PI 3.141592653589
using namespace std;

double x[3],y[3],rx,ry,r;

int trgin() {
	int i,k;
	double a,b,c,d=0;
	for (i=0; i<3; i++) {
	 k=(i+1) % 3;
	 a=sqrt((x[i]-x[k])*(x[i]-x[k])+(y[i]-y[k])*(y[i]-y[k]));
	 b=sqrt((x[i]-rx)*(x[i]-rx)+(y[i]-ry)*(y[i]-ry));
	 c=sqrt((x[k]-rx)*(x[k]-rx)+(y[k]-ry)*(y[k]-ry));
     d=d+acos((b*b+c*c-a*a)/(2*b*c));
	}
	if ((d-PI*2)<0.000001 && (d-PI*2)>-0.000001) return 1;
	return 0;
}
int line(double x1,double y1,double x2,double y2) {
    double x,y,m,d;
	if (y1==y2) { x=rx;y=y1;} else if (x1==x2) { x=x1;y=ry;}
	else {
    m=(y2-y1)/(x2-x1);
	x=(ry-y1+m*x1+rx/m)/(m+1/m);
	y=y1+m*(x-x1);
	}
	d=(rx-x)*(rx-x)+(ry-y)*(ry-y);
    if (d==r*r) return 1; else if (d<r*r) return 2;
	return 0;
}
int main() {
	int i,j,k,f;
	while(true) {
		for (i=0;i<3;i++) {
		cin >> x[i] >> y[i];
		if (x[i]==0) break;
	}
    if (i!=3) break;
	cin >> rx >> ry >> r;
    for (i=0;i<3;i++) if ((x[i]-rx)*(x[i]-rx)+(y[i]-ry)*(y[i]-ry)>r*r) break;
	if (i==3) { cout << 'b' << endl; continue;}
	for (i=0,f=0,k=0;i<3;i++) {
		j=line(x[i],y[i],x[(i+1) % 3],y[(i+1) % 3]); k+=j;
		if (j>1) break;
	}
	if (i==3) if (trgin()==1) { cout << 'a' << endl; continue;} else if (k==0) {cout << 'd' << endl; continue;}
	cout << 'c' << endl;
	}
return 0;
 }

C++: #include<iostream>
#include<cstdlib>
#include<complex>
#define EPS 1e-10
using namespace std;

typedef complex<double> Po;


Po p[3],o;

double dot(Po x, Po y){return x.real()*y.real()+x.imag()*y.imag();}

double cross(Po x, Po y){return x.real()*y.imag() - x.imag()*y.real();}

double area(Po a, Po b, Po c){return abs(cross(b-a,c-a))/2;}

bool inter(Po a, Po b, Po c, Po x){
  return (fabs(area(a,b,x)+area(b,c,x)+area(c,a,x)-area(a,b,c)) < EPS);
}

double seg_dis(Po a, Po b, Po 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);
}

int main(){
  double x[4],y[4],r;

  for(;;){
    cin >> x[0] >> y[0];
    if(!x[0] && !y[0])break;

    for(int i=1;i<4;i++)cin >> x[i] >> y[i];
    cin >> r;

    for(int i=0;i<3;i++)p[i] = Po(x[i],y[i]);
    o = Po(x[3],y[3]);


    bool f[3];
    for(int i=0;i<3;i++)f[i] = !(abs(o-p[i])>r);
    if(f[0] && f[1] && f[2])cout << "b\n";
    else{
      if(inter(p[0],p[1],p[2],o)){
	for(int i=0;i<3;i++)f[i] = !(seg_dis(p[i],p[(i+1)%3],o)<r);
	if(f[0] && f[1] && f[2])cout << "a\n";
	else cout << "c\n";
      }else{
	for(int i=0;i<3;i++)f[i] = !(seg_dis(p[i],p[(i+1)%3],o)>r);
	if(f[0] || f[1] || f[2])cout << "c\n";
	else cout << "d\n";
      }
    }
  }
}