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 <sstream> 
#include <iomanip> 
#include <algorithm> 
#include <cmath> 
#include <string> 
#include <vector> 
#include <list> 
#include <queue> 
#include <stack> 
#include <set> 
#include <map> 
#include <bitset> 
#include <numeric> 
#include <climits> 
#include <cfloat> 
using namespace std; 

class Point
{
public:
    int y, x;
    Point(){
        y = x = 0;
    }
    Point(int y0, int x0){
        y = y0;
        x = x0;
    }
    Point operator+(const Point& p) const{
        return Point(y + p.y, x + p.x);
    }
    Point operator-(const Point& p) const{
        return Point(y - p.y, x - p.x);
    }
    Point operator*(int a) const{
        return Point(y * a, x * a);
    }
    double length2() const{
        return y * y + x * x;
    }
    double dist2(const Point& p) const{
        return (y - p.y) * (y - p.y) + (x - p.x) * (x - p.x);
    }
    int dot(const Point& p) const{
        return y * p.y + x * p.x; // |a|*|b|*cosÆ
    }
    int cross(const Point& p) const{
        return x * p.y - y * p.x; // |a|*|b|*sinÆ
    }
};

int segmentCircleCollide(const Point& a1, const Point& a2, const Point& c, int r)
{
    int num = 0;
    if(a1.dist2(c) < r * r)
        ++ num;
    if(a2.dist2(c) < r * r)
        ++ num;
    if(num == 2)
        return -1;
    if(num == 1)
        return 1;
    if((a2 - a1).dot(c - a1) > 0 && (a1 - a2).dot(c - a2) > 0){
        int s = (a2 - a1).cross(c - a1);
        int t = r * r * (a2 - a1).length2();
        if(s * s == t)
            return 1;
        if(s * s < t)
            return 2;
    }
    return 0;
}

bool convexContain(const vector<Point>& cp, const Point& p)
{
    int n = cp.size();
    bool ret1 = true;
    bool ret2 = true;
    for(int i=0; i<n; ++i){
        int a = (cp[(i+1)%n] - cp[i]).cross(p - cp[i]);
        if(a < 0)
            ret1 = false;
        if(a > 0)
            ret2 = false;
    }
    return ret1 || ret2;
}

int main()
{
    for(;;){
        vector<Point> p(3);
        for(int i=0; i<3; ++i){
            cin >> p[i].x >> p[i].y;
            if(p[i].x == 0)
                return 0;
        }

        Point c;
        int r;
        cin >> c.x >> c.y >> r;

        char ret = 'd';
        int num = 0;
        for(int i=0; i<3; ++i){
            if(p[i].dist2(c) <= r * r)
                ++ num;
        }
        if(num == 3)
            ret = 'b';
        else if(num > 0)
            ret = 'c';

        if(ret == 'd'){
            if(convexContain(p, c))
                ret = 'a';
            for(int i=0; i<3; ++i){
                int tmp = segmentCircleCollide(p[i], p[(i+1)%3], c, r);
                if(ret == 'd' && tmp > 0)
                    ret = 'c';
                if(ret == 'a' && tmp == 2)
                    ret = 'c';
            }
        }

        cout << ret << endl;
    }
}