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 <bits/stdc++.h>
using namespace std;

typedef complex<double> P;

const double eps = 1e-8;

struct C {
  P p;
  double r;
  C() : p(P(0,0)), r(0) {}
  C(P p, double r) : p(p), r(r) {}
};

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

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

int ccw(P p, P a, P b) {
  a -= p;
  b -= p;
  if(cross(a,b) > eps) return 1;
  if(cross(a,b) < -eps) return -1;
  if(dot(a,b) < -eps) return 2;
  if(norm(b)-norm(a) > eps) return -2;
  return 0;
}

double getDistanceLP(P s1, P s2, P p) {
  return abs(cross(s2-s1,p-s1))/abs(s2-s1);
}

double getDistanceSP(P s1, P s2, P p) {
  if(dot(s2-s1, p-s1) < 0.0) return abs(p-s1);
  if(dot(s1-s2, p-s2) < 0.0) return abs(p-s2);
  return getDistanceLP(s1, s2, p);
}

bool involve(const vector<P> &g, const C &c) {
  int n = g.size();
  for(int i = 0; i < n; ++i) {
    P a = g[i], b = g[(i+1)%n];
    double dist = getDistanceLP(a,b,c.p);
    if(cross(b-a, c.p-a) > eps && dist - c.r > -eps) {
    } else {
      return false;
    }
  }
  return true;
}

bool involve(const C &c, const vector<P> &g) {
  int n = g.size();
  for(int i = 0; i < n; ++i) {
    P a = g[i], b = g[(i+1)%n];
    double dist = max(abs(c.p - a), abs(c.p - b));
    if(c.r - dist > -eps) {
    } else {
      return false;
    }
  }
  return true;
}

bool isIntersect(const P &s1, const P &s2, const C &c) {
  double dist = getDistanceSP(s1, s2, c.p);
  return c.r - dist > eps;
}

bool isIntersect(const vector<P> &g, const C &c) {
  int n = g.size();
  for(int i = 0; i < n; ++i) {
    P a = g[i], b = g[(i+1)%n];
    if(isIntersect(a,b,c)) return true;
  }
  return false;
}

int main() {
  vector<P> g(3);
  C c;
  while(1) {
    for(int i = 0; i < g.size(); ++i) {
      cin >> g[i].real() >> g[i].imag();
      if(g[0].real() == 0 && g[0].imag() == 0) return 0;
    }
    cin >> c.p.real() >> c.p.imag();
    cin >> c.r;
    if(involve(g,c)) cout << "a" << endl;
    else if(involve(c,g)) cout << "b" << endl;
    else if(isIntersect(g,c)) cout << "c" << endl;
    else cout << "d" << endl;
  }
  return 0;
}

C++: #include <bits/stdc++.h>
using namespace std;

const double EPS = 1e-6;

typedef complex<double> C;

double ip(C a, C b){
    return a.real() * b.real() + a.imag() * a.imag();
}

double op(C a, C b){
    return a.real() * b.imag() - a.imag() * b.real();
}

double dist_lc(C p, double a, double b, double c){
    return abs(a * p.real() + b * p.imag() + c) / abs(p);
}

int main(){
    C p1, p2, p3, pr;
    double r;
    bool flg1, flg2, flg3, flg4;
    while(true){
        int s, t;
        double x1, y1, x2, y2, x3, y3, xr, yr;
        scanf("%d%d", &s, &t);
        if(s == 0 && t == 0) break;
        p1.real(s); p1.imag(t);
        x1 = p1.real(); y1 = p1.imag();
        scanf("%d%d", &s, &t);
        p2.real(s); p2.imag(t);
        x2 = p2.real(); y2 = p2.imag();
        scanf("%d%d", &s, &t);
        p3.real(s); p3.imag(t);
        x3 = p3.real(); y3 = p3.imag();
        scanf("%d%d", &s, &t);
        pr.real(s); pr.imag(t);
        xr = s; yr = t;
        scanf(" %lf", &r);
        if(abs(p1 - pr) <= r && abs(p2 - pr) <= r && abs(p3 - pr) <= r){
            puts("b");
            continue;
        }
        double a = x1 - x2, b = y1 - y2;
        double c = x2 - x3, d = y2 - y3;
        double e = x3 - x1, f = y3 - y1;
        flg1 = abs(b * xr - a * yr - b * x1 + a * y1)/abs(p1-p2) >= r;
        flg2 = abs(d * xr - c * yr - d * x2 + c * y2)/abs(p2-p3) >= r;
        flg3 = abs(f * xr - e * yr - f * x3 + e * y3)/abs(p3-p1) >= r;
        flg4 = op(p1-pr, p2-pr) + op(p2-pr, p3-pr) + op(p3-pr, p1-pr) > EPS;
        if(flg1 && flg2 && flg3 && flg4){
            puts("a");
            continue;
        }
        flg1 = abs(pr - p1) <= r;
        flg2 = abs(pr - p2) <= r;
        flg3 = abs(pr - p3) <= r;
        if(flg1 || flg2 || flg3){
            puts("c");
            continue;
        }
        flg1 = dist_lc(pr, b, -a, -b * p1.real() + a * p1.imag()) <= r &&
               ip(p1-p2, pr-p2) >= 0 && ip(p2-p1, pr-p1) >= 0;
        flg2 = dist_lc(pr, d, -c, -d * p2.real() + b * p2.imag()) <= r &&
               ip(p2-p3, pr-p3) >= 0 && ip(p3-p2, pr-p2) >= 0;
        flg3 = dist_lc(pr, f, -e, -f * p3.real() + f * p3.imag()) <= r &&
               ip(p3-p1, pr-p1) >= 0 && ip(p1-p3, pr-p3) >= 0;
        if(flg1 || flg2 || flg3){
            puts("c");
            continue;
        }
        puts("d");
    }
    return 0;
}