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 <vector>

using namespace std;

typedef long double ld;

const ld eps = 1e-9;

struct Point { // ポイントクラス兼ベクトルクラス
    ld x, y;
    Point() {}
    Point(ld x, ld y) : x(x), y(y) {}
    Point operator+(const Point &o) const {
        return Point(x + o.x, y + o.y);
    }
    Point operator-(const Point &o) const {
        return Point(x - o.x, y - o.y);
    }
    Point operator*(ld a) const {
        return Point(a*x, a*y);
    }
    Point operator/(ld a) const {
        return Point(x/a, y/a);
    }
    ld norm() const {
        return sqrt(x*x + y*y);
    }
    Point unit() const { // 単位ベクトル
        return (*this)/norm();
    }
};
ld dot(const Point &a, const Point &b) {
    return a.x * b.x + a.y * b.y;
}
ld cross(const Point &a, const Point &b) {
   return a.x * b.y - a.y * b.x;
}
#define CCW 1
#define CW -1
#define CAB 2
#define ABC -2
#define ACB 0
int ccw(Point a, Point b, Point c) {
    // aを基準とする位置ベクトルを考える
	b = b - a; c = c - a;
	if (cross(b, c) > eps) return CCW; // 反時計周り
	if (cross(b, c) < -eps) return CW;// 時計周り
	if (dot(b, c) < 0) return CAB;     // c--a--bがこの順番に一直線上
	if (b.norm() < c.norm()) return ABC; // a--b--cがこの順番に一直線上
	return ACB;                         // a--c--bが一直線上
}
typedef Point Vector;

struct Segment {
    Point from, to;
    Segment() {}
    Segment(Point from, Point to) : from(from), to(to) {}
    bool intersects(const Segment &o) const {
        return (ccw(from, to, o.from) != ccw(from, to, o.to)) &&
                    (ccw(o.from, o.to, from) != ccw(o.from, o.to, to));
    }
    ld length() const {
        return (to-from).norm();
    }
    bool contains(const Point &p) const {
        return ccw(from, to, p) == ACB;
    }
};

struct Line : public Segment {
    Line() {}
    Line(Segment s) : Segment(s) {}
    Line(Point from, Point to) : Segment(from, to) {}
    ld distance(const Point &p) const {
        Vector v1 = p - from;
        Vector v2 = to - from;
        ld S = abs(cross(v1, v2));   // v1,v2が張る平行四辺形の面積
        return S / length();         // それを線分の長さで割る
    }
    Point projection(const Point &p) const {
        Vector v1 = p - from, v2 = to - from;
        ld l = dot(v1, v2) / v2.norm();
        return from + (v2.unit()) * l;
    }
};

struct Circle {
    Point o;
    ld r;
    Circle() {}
    Circle(Point o, ld r) : o(o), r(r) {}
    Circle(ld x, ld y, ld r) : o(Point(x, y)), r(r) {}
    bool intersects(const Segment& s) const { 
        if (contains(s.from) && contains(s.to)) return false;
        Line l(s);
        if (l.distance(o) >= r - eps) return false;
        if (contains(s.from) != contains(s.to)) return true;
        return s.contains(l.projection(o));
    }
    bool contains(const Point& p) const {
        ld dx = o.x - p.x;
        ld dy = o.y - p.y;
        return r*r + eps >= dx*dx + dy*dy;
    }
};

Point p[3];
Circle C;

bool b() {
    for (int i = 0; i < 3; i++) 
        if (!C.contains(p[i]))
            return false;
    return true;
}

bool c() {
    for (int i = 0; i < 3; i++) {
        Segment s(p[i%3], p[(i+1)%3]);
        if (C.intersects(s)) 
            return true;
    }
    return false;
}

bool a() {
    Point g = (p[0] + p[1] + p[2]) / 3.0;
    Segment s(C.o, g);
    for (int i = 0; i < 3; i++) {
        Segment edge(p[i%3], p[(i+1)%3]);
        if (s.intersects(edge)) return false;
    }
    return true;
}

int main() {
    while (cin >> p[0].x >> p[0].y, p[0].x || p[0].y) {
        cin >> p[1].x >> p[1].y;
        cin >> p[2].x >> p[2].y;
        cin >> C.o.x >> C.o.y;
        cin >> C.r;
        if (b()) {
            cout << "b" << endl;
        } else if (c()) {
            cout << "c" << endl;
        } else if (a()) {
            cout << "a" << endl;
        } else {
            cout << "d" << endl;
        }
    }
    return 0;
}