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) );
}
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;
}
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) );
	}
}
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 &p1, const Point &p2, const Circle &c)
{
	if( dist( p1, p2, c.c ) > c.r )
		return false;
	else{
		double l = dist( p1, p2 );
		if( ( dist(p1, c.c)>c.r && dist(p2,c.c)>c.r+l ) ||
			( dist(p2, c.c)>c.r && dist(p1,c.c)>c.r+l ) )
			return false;
	}
	return true;
}
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( false == 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<string>
#include<vector>
#include<iostream>
#include<algorithm>
#include<cstdio>
#include<cstdlib>
#include<sstream>
#include<functional>
#include<map>
#include<set>
#include<cmath>

#define REP(i,s,e) for(int i=int(s);i<int(e);i++)
#define rep(i,n) for(int i=0;i<int(n);i++)

using namespace std;

bool f(double x1,double y1,double x2,double y2,double x4,double y4,double r){

double p=(y2-y1)/(x2-x1);
double q=y1-y4-p*(x1-x4);
double d=p*p*q*q-(1+p*p)*(q*q-r*r);

if(x1!=x2){
if(d>=0 && (-p*q+sqrt(d))/(1+p*p)<max(x2-x4,x1-x4) && (-p*q-sqrt(d))/(1+p*p)>min(x2-x4,x1-x4))
return true;
else return false;
}
else if(abs(x1-x4)<=r && max(y1-y4,y2-y4)>sqrt(r*r-(x1-x4)*(x1-x4)) && min(y1-y4,y2-y4)<-sqrt(r*r-(x1-x4)*(x1-x4)))
return true;
else return false;
}

bool g(double x1,double y1,double x2,double y2,double x3,double y3,double x4,double y4,double r){

double a=sqrt((x1-x4)*(x1-x4)+(y1-y4)*(y1-y4));
double b=(y1-y4)/(x1-x4);
double c=(y2-y3)/(x2-x3);
double x=(y2-y4-c*(x2-x4))/(b-c);
double y=(b*(y2-y4)-b*c*(x2-x4))/(b-c);

if(x1!=x4 && x2!=x3 && b!=c && a>sqrt((x-x1+x4)*(x-x1+x4)+(y-y1+y4)*(y-y1+y4)))
return true;
else if(x1==x4 && x2!=x3 && abs(y1-y4)>abs(y1-y2+c*(x2-x4)))
return true;
else if(x1!=x4 && x2==x3 && a>sqrt((x2-x1)*(x2-x1)+(b*(x2-x4)-y1+y4)*(b*(x2-x4)-y1+y4)))
return true;
else return false;

}

int main(){

double x1,y1,x2,y2,x3,y3,x4,y4,r;

while(true){

cin >> x1 >> y1 >> x2 >> y2 >> x3 >> y3 >> x4 >> y4 >> r;

if(x1==0 && y1==0) break;


double a,b,c;
a=(x1-x4)*(x1-x4)+(y1-y4)*(y1-y4);
b=(x2-x4)*(x2-x4)+(y2-y4)*(y2-y4);
c=(x3-x4)*(x3-x4)+(y3-y4)*(y3-y4);

if(a<r*r && b<r*r && c<r*r)
cout << "b" << endl;
if(!(a<r*r && b<r*r && c<r*r) && !(a>r*r && b>r*r && c>r*r))
cout << "c" << endl;

if(a>r*r && b>r*r && c>r*r){
if(f(x1,y1,x2,y2,x4,y4,r) || f(x2,y2,x3,y3,x4,y4,r) || f(x1,y1,x3,y3,x4,y4,r))
cout << "c" << endl;
else if(g(x1,y1,x2,y2,x3,y3,x4,y4,r) || g(x2,y2,x1,y1,x3,y3,x4,y4,r) || g(x3,y3,x2,y2,x1,y1,x4,y4,r))
cout << "d" << endl;
else cout << "a" << endl;

}
}

	return 0;
}