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 <algorithm>
#include <vector>
#include <cmath>
using namespace std;
#define DEBUG(x) cerr << #x << " = " << x << endl
#define REP(i,n) for(int i=0;i<(int)(n);++i)
typedef double D;
const D EPS = 1e-8;
const D PI = M_PI;
struct P {
  D x, y;
  P() { }
  P(D x_, D y_) : x(x_), y(y_) { }
};
struct L {
  P a, b;
  L() { }
  L(P a_, P b_) : a(a_), b(b_) { }
};
struct C {
  P p; D r;
  C() { }
  C(P p_, D r_) : p(p_), r(r_) { }
};
P operator +(P a, P b) { return P(a.x + b.x, a.y + b.y); }
P operator -(P a, P b) { return P(a.x - b.x, a.y - b.y); }
P operator *(P a, D b) { return P(a.x * b, a.y * b); }
P operator /(P a, D b) { return P(a.x / b, a.y / b); }
P vec(P st, P en) { return en - st; }
D inp(P a, P b) { return a.x*b.x + a.y*b.y; }
D outp(P a, P b) { return a.x*b.y - a.y*b.x; }
D abs(P p) { return sqrt(inp(p,p)); }
int sgn(D a, D b) {
  if(a < b - EPS) return -1;
  if(a > b + EPS) return +1;
  return 0;
}
P proj(L l, P p) {
  P a = vec(l.a,l.b);
  P b = vec(l.a,p);
  D t = inp(a,b)/abs(a);
  return l.a + (a/abs(a))*t;
}
D dLP(L l, P p) {
    return abs(p - proj(l, p));
}
int ccw(P a, P b, P c) {
  b = vec(a,b); c = vec(a,c);
  if(sgn(outp(b,c),0.0) > 0) return +1;
  if(sgn(outp(b,c),0.0) < 0) return -1;
  if(sgn(inp(b,c),0.0) < 0)  return +2;
  if(sgn(abs(b),abs(c)) < 0) return -2;
  return 0;
}
bool in_triangle(vector<P> g, P p) {
  double a = 0.0, b = 0.0;
  REP(i,3) {
    a += abs(outp(g[i] - p, g[(i+1)%3] - p));
    b += outp(g[i], g[(i+1)%3]);
  }
  return sgn(abs(a), abs(b)) == 0;
}
int iCS(C c, L s) {
  if(sgn(abs(c.p - s.a), c.r) <= 0 && sgn(abs(c.p - s.b), c.r) <= 0)
    return -2;
  if(sgn(abs(c.p - s.a), c.r) <= 0 || sgn(abs(c.p - s.b), c.r) <= 0)
    return -1;
  if(sgn(dLP(s, c.p), c.r) < 0)
    return +2;
  if(sgn(dLP(s, c.p), c.r) == 0)
    return +1;
  return 0;
}
int main() {
  while(true) {
    int x, y;
    vector<P> tri;
    cin >> x >> y;
    if(x == 0 && y == 0) break;
    tri.emplace_back(x, y);
    cin >> x >> y;
    tri.emplace_back(x, y);
    cin >> x >> y;
    tri.emplace_back(x, y);
    C cir;
    cin >> cir.p.x >> cir.p.y >> cir.r;
    // cerr << "circle(" << cir.p.x << "," << cir.p.y << "," << cir.r << ");" << endl;
    // cerr << "line(" << tri[0].x << "," << tri[0].y << "," << tri[1].x << "," << tri[1].y << ");" << endl;
    // cerr << "line(" << tri[1].x << "," << tri[1].y << "," << tri[2].x << "," << tri[2].y << ");" << endl;
    // cerr << "line(" << tri[2].x << "," << tri[2].y << "," << tri[0].x << "," << tri[0].y << ");" << endl;
    // b
    if(sgn(abs(vec(cir.p, tri[0])), cir.r) <= 0 &&
       sgn(abs(vec(cir.p, tri[1])), cir.r) <= 0 &&
       sgn(abs(vec(cir.p, tri[2])), cir.r) <= 0) {
      cout << "b" << endl;
      continue;
    }
    // a
    if(in_triangle(tri, cir.p) &&
       sgn(cir.r, dLP(L(tri[0], tri[1]), cir.p)) <= 0 &&
       sgn(cir.r, dLP(L(tri[1], tri[2]), cir.p)) <= 0 &&
       sgn(cir.r, dLP(L(tri[2], tri[0]), cir.p)) <= 0) {
      cout << "a" << endl;
      continue;
    }
    if(iCS(cir, L(tri[0], tri[1])) == 0 &&
       iCS(cir, L(tri[1], tri[2])) == 0 &&
       iCS(cir, L(tri[2], tri[0])) == 0) {
      cout << "d" << endl;
      continue;
    }
    cout << "c" << endl;
  }
}

C++: #include <iostream>
#include <cmath>
using namespace std;

class Vector2
{
public:
	double length() const
	{
		return sqrt( static_cast<double>( x * x + y * y ) );
	}
	int clossProduct( const Vector2& v ) const
	{
		return this->x * v.y - this->y * v.x;
	}
	Vector2 operator-( const Vector2& v ) const
	{
		Vector2 ret;
		ret.x = this->x - v.x;
		ret.y = this->y - v.y;
		return ret;
	}

	int x, y;
};

int main()
{
	Vector2 tri[ 3 ];
	Vector2 cir;
	int cirR;

	while( cin >> tri[ 0 ].x >> tri[ 0 ].y && !( !tri[ 0 ].x && !tri[ 0 ].y ) ){
		for( int i = 1; i < 3; ++i ){
			cin >> tri[ i ].x >> tri[ i ].y;
		}
		cin >> cir.x >> cir.y;
		cin >> cirR;

		int flags = 0;
		for( int i = 0; i < 3; ++i ){
			Vector2 ao = cir - tri[ i ];
			Vector2 ab = tri[ ( i + 1 ) % 3 ] - tri[ i ];
			double distance = abs( ab.clossProduct( ao ) ) / ab.length();
			if( distance >= cirR ){
				flags |= ( 1 << i );
			}
		}

		char result;
		if( flags == 7 ){
			int dir = 0;
			for( int i = 0; i < 3; ++i ){
				Vector2 oa = tri[ i ] - cir;
				Vector2 ob = tri[ ( i + 1 ) % 3 ] - cir;
				if( oa.x * ob.y > oa.y * ob.x ){
					dir |= ( 1 << i );
				}
			}
			if( dir % 7 == 0 ){
				result = 'a';
			}else{
				result = 'd';
			}
		}else if( flags == 0 ){
			int dir = 0;
			for( int i = 0; i < 3; ++i ){
				Vector2 oa = tri[ i ] - cir;
				if( oa.length() <= cirR ){
					dir |= ( 1 << i );
				}
			}
			if( dir == 7 ){
				result = 'b';
			}else{
				return 'c';
			}
		}else{
			result = 'c';
		}

		cout << result << endl;
	}

	return 0;
}