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 <cstdio>
#include <cstdlib>
#include <cmath>
#include <climits>
#include <cfloat>
#include <map>
#include <utility>
#include <set>
#include <iostream>
#include <memory>
#include <string>
#include <vector>
#include <algorithm>
#include <functional>
#include <sstream>
#include <complex>
#include <stack>
#include <queue>
typedef long long ll;
using namespace std;
 
struct Pos{
    double x,y;
    Pos(double x,double y) : x(x) , y(y) {}
    Pos(){}
};
 
struct Triangle{
    Pos pos[3];
};
struct Circle{
    Pos pos;
    double r;
};
 
static const double kyuriEPS = 1e-1;
typedef std::complex<double> myPoint;
double dot(const myPoint& a, const myPoint& b) {return real(conj(a)*b);}
struct myL : public std::vector<myPoint> {
    myL(const myPoint &a, const myPoint &b) {push_back(a); push_back(b);}
};
 
bool intersectSP(const myL &s, const myPoint &p) {
    return abs(s[0]-p)+abs(s[1]-p)-abs(s[1]-s[0]) < kyuriEPS;
} 
myPoint projection(const myL &l, const myPoint &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 distanceSP(const myL &s, const myPoint &p) {
    const myPoint r = projection(s, p);
    if (intersectSP(s, r)) return abs(r - p);
    return std::min(abs(s[0] - p), abs(s[1] - p));
}
//ct
int judgeHit(const Circle &c,const Triangle &t){
    int counter = 0 , ctt = 0;
    for(int i = 0 ; i < 3 ; i++){
        if( abs(myPoint(c.pos.x,c.pos.y)-myPoint(t.pos[i].x,t.pos[i].y)) <= c.r + kyuriEPS ){
            ctt++;
        }
    }
    if( ctt == 3 ) return 1;
	if( ctt >= 1 ) return 2;
    for(int i = 0 ; i < 3 ; i++){
        double dist = distanceSP(myL(myPoint(t.pos[i].x,t.pos[i].y),myPoint(t.pos[(i+1)%3].x,t.pos[(i+1)%3].y)),myPoint(c.pos.x,c.pos.y));
        if( dist <= c.r + kyuriEPS ){
            counter++;
        }
    }
    if( counter >= 1 ) return 2;
     
	double tS2 = fabs( (t.pos[1].x-t.pos[0].x)*(t.pos[2].y-t.pos[0].y) - (t.pos[1].y-t.pos[0].y)*(t.pos[2].x-t.pos[0].x) );
	
	// a.pos[i] on B
	double area = 0;
	for(int j = 0 ; j < 3 ; j++){
		area += fabs( (t.pos[j].x-c.pos.x)*(t.pos[(j+1)%3].y-c.pos.y) - (t.pos[j].y-c.pos.y)*(t.pos[(j+1)%3].x-c.pos.x) );
	}
	//std::cout << area << " " << tS2 << std::endl;
	
	if( fabs( fabs(area) - tS2 ) == 0 ) return 0; 	
	
	return 3;
}
 
int main(){
    Triangle t;
    Circle c;
    while(cin >> t.pos[0].x >> t.pos[0].y && (t.pos[0].x||t.pos[0].y)){
        for(int i = 1 ; i < 3 ; i++)
            cin >> t.pos[i].x >> t.pos[i].y;
        cin >> c.pos.x >> c.pos.y >> c.r;
        cout << (char)('a'+judgeHit(c,t)) << endl;
    }
}

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

const double eps = 1e-8;
double x[4], y[4], r;

double distSq(double x1, double y1, double x2, double y2) {
	return (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1);
}

double perpDist(double x1, double y1, double x2, double y2, double cx, double cy) {
	double a, b, x, y;
	double res = min(distSq(x1, y1, cx, cy), distSq(x2, y2, cx, cy));
	if (x1 == x2) {
		if (min(y1, y2) <= cy && cy <= max(y1, y2)) {
			res = (cx - x1) * (cx - x1);
		}
	} else {
		a = (y2 - y1) / (x2 - x1);
		b = y1 - a * x1;
		x = (cx + a * (cy - b)) / (1 + a * a);
		y = a * x + b;
		if (min(x1, x2) <= x && x <= max(x1, x2) && min(y1, y2) <= y && y <= max(y1, y2)) {
			res = distSq(x, y, cx, cy);
		}
	}
	return res;
}

bool A() {
	int c = 0;
	for (int i=0; i<3; ++i) {
		if (0 <= (x[(i+1)%3] - x[i]) * (y[3] - y[i]) - (y[(i+1)%3] - y[i]) * (x[3] - x[i])) {
			++ c;
		}
	}
	if (c % 3 != 0) return false;
	for (int i=0; i<3; ++i) {
		if (perpDist(x[i], y[i], x[(i+1)%3], y[(i+1)%3], x[3], y[3]) + eps < r * r) {
			return false;
		}
	}
	return true;
}

bool B() {
	for (int i=0; i<3; ++i) {
		if (r * r < distSq(x[i], y[i], x[3], y[3])) {
			return false;
		}
	}
	return true;
}

bool C() {
	for (int i=0; i<3; ++i) {
		if (perpDist(x[i], y[i], x[(i+1)%3], y[(i+1)%3], x[3], y[3]) <= r * r + eps) {
			return true;
		}
	}
	return false;
}

int main() {
	while (cin >> x[0] >> y[0], (x[0] || y[0])) {
		for (int i=1; i<4; ++i) cin >> x[i] >> y[i];
		cin >> r;
		if (B()) {
			cout << "b" << endl;
		} else if (A()) {
			cout << "a" << endl;
		} else if (C()) {
			cout << "c" << endl;
		} else {
			cout << "d" << endl;
		}
	}
	return 0;
}