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 <cstdio>
#include <cstdlib>
#include <cmath>
#include <ctime>
#include <cstring>
#include <climits>
#include <algorithm>
#include <map>
#include <set>
#include <stack>
#include <vector>
#include <queue>
#include <bitset>
#include <string>
#include <complex>
using namespace std;
 
#define rep(i,n) for(int i=0; i<n; i++)
#define repa(i,s,e) for(int i=s; i<=e; i++)
#define repd(i,s,e) for(int i=s; i>=e; i--)
 
typedef long long ll;
typedef vector<int> vi;
typedef pair<int, int> pii;

// geometry library

const double EPS = 1.0e-8;
const double INF = 1.0e12;
typedef complex<double> P;
namespace std {
	bool operator<(const P& p, const P& q) {
		return p.real() != q.real() ? p.real() < q.real() : p.imag() < q.imag(); 
	}
}

double cross(const P p, const P q) {
	return imag(conj(p) * q); 
}

double dot(const P p, const P q) {
	return real(conj(p) * q);
}

class L : public vector<P> {
public:
	L(const P& p, const P& q) {
		this->push_back(p);
		this->push_back(q);
	}
};

class C {
public:
	P p;
	double r;
	C() : p(0.0, 0.0), r(0.0) {}
	C(const P& p_, double r_) : p(p_), r(r_) {} 
};

// compute distance between 2 points
double distP2P(const P& p, const P& q) {
	P d = p - q;
	return sqrt(norm(d));
}

// compute shortest distance to line
double distP2L(const P& p, const L& l) {
	double a = l[1].real() - l[0].real();
	double b = l[0].imag() - l[1].imag();
	double c = - a * l[0].real() - b * l[0].imag();
	double de = abs(a * p.real() + b * p.imag() + c);
	double nm = sqrt(a * a + b * b);
	if(nm == 0.0) return min(distP2P(p, l[0]), distP2P(p, l[1]));
	return de / nm;
}

// compute shortest distance to line segment
double distP2LS(const P& p, const L& l) {
	double d1 = dot(p - l[0], l[1] - l[0]);
	double d2 = dot(p - l[1], l[0] - l[1]);
	if(d1 < 0 || d2 < 0) {
		return min(distP2P(p, l[0]), distP2P(p, l[1]));
	}
	return distP2L(p, l);
}

// code for problem
int x, y, r;
P tr[3];
C cc;

bool isPinT(const P& p, P t[3]) {
	P e0 = t[0] - p;
	P e1 = t[1] - p;
	P e2 = t[2] - p;
	double d0 = cross(e0, e1);
	double d1 = cross(e1, e2);
	double d2 = cross(e2, e0);
	if(d0 * d1 > 0.0 && d1 * d2 > 0.0 && d2 * d0 > 0.0) return true;
	return false;
}

void solve() {
	// triangle is in circle ?
	vector<double> d(3);
	d[0] = distP2P(tr[0], cc.p);
	d[1] = distP2P(tr[1], cc.p);
	d[2] = distP2P(tr[2], cc.p);
	if(d[0] <= cc.r && d[1] <= cc.r && d[2] <= cc.r) {
		printf("b\n");
		return;
	}

	// circle is in triangle ?
	bool in = isPinT(cc.p, tr);
	L l0(tr[0], tr[1]);
	L l1(tr[1], tr[2]);
	L l2(tr[2], tr[0]);
	d[0] = distP2LS(cc.p, l0);
	d[1] = distP2LS(cc.p, l1);
	d[2] = distP2LS(cc.p, l2);
	if(in && d[0] >= cc.r && d[1] >= cc.r && d[2] >= cc.r) {
		printf("a\n");
		return;
	}

	// not intersect ?
	sort(d.begin(), d.end());
	if(!in && d[0] > r) {
		printf("d\n");
		return;
	}

	// intersects each other ?
	printf("c\n");
}

int main() {
	while( 1 ) {
		scanf("%d%d", &x, &y);
		if(x == 0 && y == 0) break;
		tr[0] = P(x, y);

		scanf("%d%d", &x, &y);
		tr[1] = P(x, y);

		scanf("%d%d", &x, &y);
		tr[2] = P(x, y);

		scanf("%d%d%d", &x, &y, &r);
		P p = P(x, y);
		cc = C(p, r);

		solve();
	}
}

C++: #include <vector>
#include <iostream>
#include <algorithm>
using namespace std;
int main() {
	while (true) {
		bool fin = false;
		vector<int> x(3), y(3);
		for (int i = 0; i < 3; ++i) {
			cin >> x[i] >> y[i];
			if (i == 0 && x[i] == 0 && y[i] == 0) {
				fin = true;
				break;
			}
		}
		if (fin) break;
		int xc, yc, r;
		cin >> xc >> yc >> r;
		vector<int> crs(3);
		for (int i = 0; i < 3; ++i) {
			int vx = xc - x[i], vy = yc - y[i];
			int wx = x[(i + 1) % 3] - x[i], wy = y[(i + 1) % 3] - y[i];
			crs[i] = vx * wy - wx * vy;
		}
		bool inner = true;
		for (int i = 0; i < 3; ++i) {
			if (1LL * crs[i] * crs[(i + 1) % 3] < 0) {
				inner = false;
			}
		}
		int contain = 0;
		for (int i = 0; i < 3; ++i) {
			int dx = x[i] - xc, dy = y[i] - yc;
			if (dx * dx + dy * dy <= r * r) {
				++contain;
			}
		}
		int intersect = 0, border = 0;
		for (int i = 0; i < 3; ++i) {
			int dx = x[(i + 1) % 3] - x[i], dy = y[(i + 1) % 3] - y[i];
			int px = x[i] - xc, py = y[i] - yc;
			int qx = x[(i + 1) % 3] - xc, qy = y[(i + 1) % 3] - yc;
			if (px * px + py * py <= r * r || qx * qx + qy * qy <= r * r) {
				++intersect;
			}
			else {
				int pdot = dx * px + dy * py;
				int qdot = dx * qx + dy * qy;
				if (pdot <= 0 && qdot >= 0) {
					long long ca = 1LL * crs[i] * crs[i];
					long long cb = 1LL * (dx * dx + dy * dy) * (r * r);
					if (ca <= cb) {
						++intersect;
					}
					if (ca == cb) ++border;
				}
			}
		}
		if (inner) {
			if (contain == 3) {
				cout << "b" << endl;
			}
			else if (intersect - border == 0) {
				cout << "a" << endl;
			}
			else {
				cout << "c" << endl;
			}
		}
		else {
			if (contain == 3) {
				cout << "b" << endl;
			}
			else if (intersect == 0) {
				cout << "d" << endl;
			}
			else {
				cout << "c" << endl;
			}
		}
	}
	return 0;
}