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<cmath>
#include<cstdlib>

using namespace std;

struct Point{ int x,y; };
struct Triangle{ Point a,b,c; };
struct Circle{ Point c; int r; };

inline double dist(const Point &p1, const Point &p2){
	return sqrt( pow(p2.x-p1.x,2.0)+pow(p2.y-p1.y,2.0) );
}
double dist(const Point &p1, const Point &p2, const Point &p)
{
	if( p1.x == p2.x ){
		return abs( p1.x - p.x );
	}else{
		double a = -(p2.y-p1.y)/double(p2.x-p1.x);
		double b = 1;
		double c = -a * p1.x - p1.y;

		return fabs( a * p.x + b * p.y + c ) / sqrt( pow(a,2.0)+pow(b,2.0) );
	}
}
inline bool isLeft(const Point &a, const Point &b, const Point &p){
	double vx = b.x - a.x, vy = b.y - a.y;
	double px = p.x - a.x, py = p.y - a.y;
	return (vx * py - px * vy) > 0;
}

inline bool isInside(const Point &p, const Triangle &t )
{
	if( !isLeft( t.a, t.b, p ) )
		return false;
	if( !isLeft( t.b, t.c, p ) )
		return false;
	if( !isLeft( t.c, t.a, p ) )
		return false;

	return true;
}

bool Contains(const Triangle &t, const Circle &c)
{
	if( isInside( c.c, t ) ){
		if( c.r < dist( t.a, t.b, c.c ) &&
			c.r < dist( t.b, t.c, c.c ) &&
			c.r < dist( t.c, t.a, c.c ) )
			return true;
	}
	return false;
}

bool IsContained(const Triangle &t, const Circle &c)
{
  if( dist( t.a, c.c ) < c.r &&
	  dist( t.b, c.c ) < c.r &&
	  dist( t.c, c.c ) < c.r )
	  return true;
  else
	  return false;
}
bool isCross(const Point &tp1, const Point &tp2, const Circle &c)
{
	Point p1 = tp1;
	Point p2 = tp2;
	if( p2.x < p1.x )
		swap( p1, p2 );

	if( dist( p1, p2, c.c ) > c.r )
		return false;
	else{
		if( p1.x != p2.x ){
			double x1, x2;
			double a = (p2.y - p1.y) / (p2.x - p1.x);
			double b = -a * p1.x + p1.y;
			double A = 1 + pow(a,2.0);
			double B = -( c.c.x - a * c.c.x + a * c.c.y );
			double C = pow( c.c.x, 2.0 ) + pow( b, 2.0 ) - 2*b*c.c.y + pow( c.c.y, 2.0 ) - pow( c.r,2.0 );

			x1 = ( -B + sqrt( pow(B,2.0) - A * C ) ) / A;
			x2 = ( -B - sqrt( pow(B,2.0) - A * C ) ) / A;

			if( p1.x <= x1 && x1 <= p2.x || p1.x <= x2 && x2 <= p2.x )
				return true;
			else
				return false;
		}else{
			if( p2.y < p1.y )
				swap( p1, p2 );

			double d = dist( p1, p2, c.c );
			double l = sqrt( pow(c.r,2.0) - pow(d,2.0) );

			double y1 = c.c.y - l;
			double y2 = c.c.y + l;

			if( p1.y <= y1 && y1 <= p2.y || p1.y <= y2 && y2 <= p2.y )
				return true;
			else
				return false;
		}
	}
}
bool isCross(const Triangle &t, const Circle &c ){

	if( !isCross(t.a,t.b,c) && !isCross(t.b,t.c,c) && !isCross(t.c,t.a,c) )
		return false;
	else
		return true;
}

char judgeRel(const Triangle &t, const Circle &c){
	if( Contains( t,c ) )
		return 'a';
	if( IsContained( t, c ) )
		return 'b';
	if( !isCross( t,c ) )
		return 'd';
	else
		return 'c';
}

int main()
{
	while(true){
		Triangle t;
		Circle c;
		cin >> t.a.x >> t.a.y;
		if( t.a.x == 0 && t.a.y == 0 )
			break;
		cin >> t.b.x >> t.b.y >> t.c.x >> t.c.y >> c.c.x >> c.c.y >> c.r;
		cout << judgeRel( t, c ) << endl;
	}
	return 0;
}

C++: #include <iostream>
#include <cmath>
#include <algorithm>
#include <complex>
#include <vector>
using namespace std;
typedef long double Double;
typedef complex<Double> P;
#define Y imag()
#define X real()
const Double EPS = (long double) 1 * 1e-12;

struct L : public vector<P> {L(const P &a, const P &b) {push_back(a); push_back(b);} };

Double get(P a[3]){
	Double A = abs(a[0]-a[1]);
	Double B = abs(a[1]-a[2]);
	Double C = abs(a[2]-a[0]);
	Double s = (A+B+C) / 2.0;
	Double S = sqrt( s * (s-A) * (s-B) * (s-C) );
	return S;
}
Double nijo(P a){
	return a.X * a.X + a.Y * a.Y;
}
 
Double dot(const P& a, const P& b) {return real(conj(a)*b);}
P projection(const L &l, const P &p) {Double t = dot(p-l[0], l[0]-l[1]) / norm(l[0]-l[1]);return l[0] + t*(l[0]-l[1]);}

Double getPedal(const L &l, const P &p) {
	return abs(p - projection(l, p));
}

int main(){
	P a[3],c;Double r;
	while(cin >> a[0].X >> a[0].Y && (a[0].X != 0 || a[0].Y != 0)){
		for(int i = 1 ; i < 3 ; i++) cin >> a[i].X >> a[i].Y;
		cin >> c.X >> c.Y >> r;
		Double S = get(a);
		for(int i = 0 ; i < 3 ; i++){
			P b[3];
			b[0] = a[i];
			b[1] = a[(i+1)%3];
			b[2] = c;
			S -= get(b);
		}
		bool in = abs(S) < EPS;
		bool hit = getPedal(L(a[0],a[1]),c) < r - EPS || getPedal(L(a[1],a[2]),c) < r - EPS || getPedal(L(a[0],a[2]),c) < r - EPS;
		//cout << getPedal(L(a[0],a[1]),c) << " " << r << endl;
		if(  nijo(a[0]-c) < r*r && nijo(a[1]-c) < r*r && nijo(a[2]-c) < r*r) cout << "b" << endl;
		else if(in && !hit) cout << "a" << endl;
		else if(hit) cout << "c" << endl;
		else cout << "d" << endl;
	}
}