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;

#define FOR(i,a,b) for(int i=(a);i<(b);i++)
#define REP(i,n) FOR(i,0,n)
#define EPS (1e-8)

typedef complex<double> P;
typedef P Vec;

struct Line {
	P p;
	Vec v;
	
	Line() {}
	Line(P p, Vec v) : p(p), v(v) {}
	
	P getPoint(double t) {
		return p + v * t;
	}
};

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

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

bool isInCir(Circle &c, P &p) { return norm(c.p - p) <= c.r*c.r + EPS; }

bool isInSeg(Line &s, P &p) {
	Vec a = s.v, b = p - s.p;
	double l1 = abs(a), l2 = abs(b);
	
	return dot(a, b) >= l1*l2 - EPS && dot(a, b) <= l1*l2 + EPS && l1 >= l2 - EPS;
}

bool isInTri(P &a, P &b, P &c, P &d) {
	P p[4];
	p[0] = a; p[1] = b; p[2] = c; p[3] = d;
	
	int cnt = 0;
	REP(i, 3) {
		Vec v = p[3] - p[i];
		Line s(p[i], p[(i + 1) % 3] - p[i]);
		
		if (isInSeg(s, p[3])) return true;
		if (cross(s.v, v) < -EPS) cnt++;
	}
	return (cnt == 0 || cnt == 3);
}

double distanceSP(Line &s, P &p) {
	Vec a = p - s.p, b = p - s.getPoint(1.0);
	
	if (dot(s.v, a) < -EPS) return abs(a);
	if (dot(s.v, a) > dot(s.v, s.v) + EPS) return abs(b);
	return abs(cross(s.v, a)) / abs(s.v);
}

int main() {
	double x1, y1, x2, y2, x3, y3, x4, y4, r;
	while (cin >> x1 >> y1, x1 || y1) {
		cin >> x2 >> y2 >> x3 >> y3 >> x4 >> y4 >> r;
		
		P p[4] = {P(x1, y1), P(x2, y2), P(x3, y3), P(x4, y4)};
		Line seg[3] = {Line(p[0], p[1] - p[0]), Line(p[1], p[2] - p[1]), Line(p[2], p[0] - p[2])};
		Circle c(p[3], r);		
		
		double m = min(distanceSP(seg[0], c.p), min(distanceSP(seg[1], c.p), distanceSP(seg[2], c.p)));
		
		if (isInCir(c, p[0]) && isInCir(c, p[1]) && isInCir(c, p[2]))
			puts("b");
		else if (isInTri(p[0], p[1], p[2], p[3])) {
			if (c.r <= m + EPS) puts("a");
			else puts("c");
		}
		else if (c.r < m - EPS) puts("d");
		else puts("c");
	}
	return 0;
}

C++: #include <iostream>
#include <algorithm>
#include <complex>
#include <vector>
#include <cmath>

using namespace std;

#define F first
#define S second

#define X real()
#define Y imag()

#define EPS (1e-7)

typedef complex<double> Point;
typedef pair<double, Point> Circle;

bool isPointInCircle(Point p, Circle c) {
  if(abs(p-c.S) < abs(c.F)) return true;
  return false;
}

double dot(Point a, Point b) {
  return a.X*b.X+a.Y*b.Y;
}

double cross(Point a, Point b) {
  return a.X*b.Y-a.Y*b.X;
}

typedef pair<Point, Point> Segment;
typedef Segment Line;

double distanceLP(Line l, Point p) {
  return abs(cross(l.S-l.F, p-l.F)) / abs(l.S-l.F);
}

double distanceSP(Segment s, Point p) {
  Point a = s.F, b = s.S;
  
  if(dot(b-a, p-a) < EPS) return abs(p-a);
  if(dot(a-b, p-b) < EPS) return abs(p-b);
  return distanceLP(s, p);
}

int main() {
  
  while(1) {
    Point points[3];
    Circle circle;
  
    for(int i=0; i<3; i++) {
      double x, y; cin >> x >> y;
      if(x == 0) return 0;
    
      points[i] = Point(x, y);
    }
  
    {
      double x, y; cin >> x >> y;
      circle.S = Point(x, y);
      cin >> circle.F;
    }
  
    bool ok = 1;
    for(int i=0; i<3; i++) {
      ok &= isPointInCircle(points[i], circle);
    }
    if(ok) {
      cout << 'b' << endl;
      continue;
    }
    
    int ins = 0;
    for(int i=0; i<3; i++) {
      Segment seg = make_pair(points[(i+1)%3], points[i]);
      ins += circle.F < distanceSP(seg, circle.S) + EPS;
    }
    if(ins == 3) {
      int flg = 0;
      for(int i=0; i<3; i++) {
        if(dot(points[(i+1)%3]-points[i], circle.S-points[i]) * dot(points[(i+2)%3]-points[(i+1)%3], circle.S-points[(i+1)%3]) > 0) {
          flg ++;
        }
      }
      if(flg == 3) cout << 'a' << endl;
      else cout << 'd' << endl;
      continue;
    }
    
    cout << 'c' << endl;
  }
  
  return 0;
}