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 <bits/stdc++.h>
 
using namespace std;
 
#define EPS 1e-7
 
class Point {
  public:
    double x,y;
    Point(double x = 0.0, double y = 0.0) : x(x), y(y) {}
    Point operator + (Point p) { return Point(x+p.x, y+p.y); }
    Point operator - (Point p) { return Point(x-p.x, y-p.y); }
    Point operator * (double a) { return Point(x*a, y*a); }
    Point operator / (double a) { return Point(x/a, y/a); }
    double abs() { return sqrt(norm()); }
    double norm() { return x * x + y * y; }
};
 
double dot(Point a, Point b)
{
    return a.x*b.x + a.y*b.y;
}
double cross(Point a, Point b)
{
    return a.x*b.y - a.y*b.x;
}
 
class Segment {
  public:
    Point t, s;
    Segment(Point t, Point s) : t(t), s(s) {}
};
 
class Circle {
  public:
    double x, y, r;
    Circle(double x = 0.0, double y = 0.0, double r = 0.0) : x(x), y(y), r(r) {}
};
 
Point point[3];
Circle ci;
 
int ccw(Point p0, Point p1, Point p2)
{
    Point a = p1 - p0;
    Point b = p2 - p0;
    if (cross(a, b) > EPS) return 1;
    if (cross(a, b) < -EPS) return -1;
    if (dot(a, b) < -EPS) return 2;
    if (a.norm() < b.norm()) return -2;
    return 0;
}
 
bool check()
{
    Point cp = Point(ci.x, ci.y);
    int c1 = ccw(point[0], point[1], cp);
    int c2 = ccw(point[1], point[2], cp);
    int c3 = ccw(point[2], point[0], cp);
    return (c1 == c2 && c2 == c3);
}
 
typedef Segment Line;

double getDistanceLP(Line l, Point p)
{
    return abs(cross(l.s-l.t, p-l.t) / (l.s-l.t).abs());
}
 
double getDistanceSP(Segment seg, Point p)
{
    if (dot(seg.s-seg.t, p-seg.t) < 0.0) return (p-seg.t).abs();
    if (dot(seg.t-seg.s, p-seg.s) < 0.0) return (p-seg.s).abs(); 
    return getDistanceLP(seg, p);
}
 
int getStateCS(Circle c, Segment seg)
{
    Point cp = Point(c.x, c.y);
    if (getDistanceSP(seg, cp) >= c.r) {
	return 1;
    }
    return 3;
}
 
bool isCross()
{
    Point cp = Point(ci.x, ci.y);
 
    Segment seg[3] = {
	Segment(point[0], point[1]),
	Segment(point[1], point[2]),
	Segment(point[2], point[0])
    };
    int cnt = 0;
    for (int i = 0; i < 3; i++) {
	if (getDistanceSP(seg[i], cp) > ci.r) {
	    cnt++;
	}
    }
 
    if (cnt == 3 && !check()) return false;
    return true;
}
 
void judge()
{
    Segment seg[3] = {
	Segment(point[0], point[1]),
	Segment(point[1], point[2]),
	Segment(point[2], point[0])
    };
    Point cp = Point(ci.x, ci.y);
    int TinC = 0;
    for (int i = 0; i < 3; i++, TinC++) {
	if ((point[i]-cp).abs() > ci.r) {
	    break;
	}
    }
 
    if (TinC == 3) { 
	cout << 'b' << endl;
	return;
    }
 
    int CinT = 0;
    for (int i = 0; i < 3; i++) {
	if (getStateCS(ci, seg[i]) == 1) {
	    CinT++;
	}
    }
    if (CinT == 3 && check()) {
	cout << 'a' << endl;
    } else {
	if (isCross()) {
	    cout << 'c' << endl;
	} else {
	    cout << 'd' << endl;
	}
    }
}
 
int main()
{
    while (1) {
	cin >> point[0].x >> point[0].y;
	if (!point[0].x && !point[0].y) break;
	for (int i = 0; i < 2; i++) {
	    cin >> point[i+1].x >> point[i+1].y;
	}
	cin >> ci.x >> ci.y >> ci.r;
	judge();
    }
    return 0;
}