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 <cstdio>
#include <cassert>
#include <cstring>
#include <vector>
#include <valarray>
#include <array>
#include <queue>
#include <set>
#include <unordered_set>
#include <map>
#include <unordered_map>
#include <algorithm>
#include <cmath>
#include <complex>
#include <random>

using namespace std;
typedef long long ll;
typedef unsigned long long ull;

typedef long double R;
typedef complex<R> P;

const R EPS = 1e-10;

int sgn(R a) {
    if (a < -EPS) return -1;
    if (a > EPS) return 1;
    return 0;
}

int sgn(R a, R b) {
    return sgn(b-a);
}

bool near(P a, P b) {
    return !sgn(abs(a-b));
}

R cross(P a, P b) { return a.real()*b.imag() - a.imag()*b.real(); }
R dot(P a, P b) { return a.real()*b.real() + a.imag()*b.imag(); }

int ccw(P a, P b, P c) {
    if (near(a, c) || near(b, c)) return 0;
    int s = sgn(cross(b-a, c-a));
    if (s) return s;
    if (dot(b-a, c-a) < 0) return 2;
    if (dot(a-b, c-b) < 0) return -2;
    return 0;
}

struct L {
    P x, y;
    L() {};
    L(P x, P y) : x(x), y(y) {};
};

P vec(const L &l) {
    return l.y-l.x;
}

R distLP(const L &l, const P &p) {
    return abs(cross(vec(l), p-l.x)/abs(vec(l)));
}

R distSP(const L &s, const P &p) {
    P s2 = vec(s)*P(0, 1);
    if (ccw(s.x, s.x+s2, p) == 1) return abs(s.x-p);
    if (ccw(s.y, s.y+s2, p) == -1) return abs(s.y-p);
    return min(min(abs(s.x-p), abs(s.y-p)), distLP(s, p));
}

struct C {
    P p;
    R r;
    C() {}
    C(P p, R r) : p(p), r(r) {}
};

struct T {
    P d[3];
    T() {}
    T(P x, P y, P z) {
        d[0] = x; d[1] = y; d[2] = z;
    }
    P& operator[](int p) {
        return d[p];
    }
    const P& operator[](int p) const {
        return d[p];
    }
    int size() const {
        return 3;
    }
};

P cu(const T &t, int i) {
    return t[(i%3+3)%3];
}

//0:P is out 1:P is on line 2:P is in
int contains(const T &pol, P p) {
    int in = -1;
    for (int i = 0; i < (int)pol.size(); i++) {
        P a=cu(pol,i)-p, b=cu(pol,i+1)-p;
        if (ccw(a, b, P(0, 0)) == 0) return 1;
        if (imag(a) > imag(b)) swap(a, b);
        if (imag(a) <= 0 && 0 < imag(b)) {
            if (cross(a, b) < 0) in *= -1;
        }
    }
    return in+1;
}

bool solve() {
    T t;
    for (int i = 0; i < 3; i++) {
        double x, y;
        if (scanf("%lf %lf", &x, &y) != 2) return false;
        t[i] = P(x, y);
    }
    C c;
    double x, y;
    cin >> x >> y;
    c.p = P(x, y);
    double r;
    cin >> r;
    c.r = r;

    R ma = -1e9, mi = 1e9;
    for (int i = 0; i < 3; i++) {
        ma = max(ma, abs(t[i]-c.p));
        mi = min(mi, distSP(L(cu(t, i), cu(t, i+1)), c.p));
    }
//    cout << ma << " " << mi << endl;
    if (sgn(ma, c.r) != -1) {
        cout << "b" << endl;
        return true;
    }

    if (contains(t, c.p) != 0 && sgn(c.r, mi) != -1) {
        cout << "a" << endl;
        return true;
    }

    if (sgn(mi, c.r) != -1) {
        cout << "c" << endl;
        return true;
    }
    cout << "d" << endl;
    return true;
}

int main() {
    while (solve()) {}
    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);
	}
	if (((x>=x1 && x<=x2) || (x>=x2 && x<=x1)) && ((y>=y1 && y<=y2) || (y>=y2 && y<=y1))){
	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;
 }