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 <cstdio>
#include <cmath>
#include <vector>
#include <map>
#include <stack>
#include <queue>
#include <algorithm>
#include <set>

#define FOR(i,a,b) for(int i=(a);i<(b);i++)
#define REP(i,j) FOR(i,0,j)
#define mp std::make_pair

typedef long long ll;
typedef unsigned long long ull;

const int INF = 1001001001;

// S N E W(南北東西)
const int dx[8] = {0, 0, 1, -1, 1, 1, -1, -1}, dy[8] = {1, -1, 0, 0, 1, -1, 1, -1};

struct P{
    P(){}
    P(ll _r, ll _i):r(_r), i(_i){}
    void real(const ll& v){r = v;}
    void imag(const ll& v){i = v;}
    ll r, i;
};

ll real(const P& p){return p.r;}
ll imag(const P& p){return p.i;}
ll norm(const P& p){ll r = real(p), i = imag(p); return r * r + i * i;}

P operator+(const P& lhs, const P& rhs){
    return P(real(lhs)+real(rhs), imag(lhs)+imag(rhs));
}
P operator-(const P& lhs, const P& rhs){
    return P(real(lhs)-real(rhs), imag(lhs)-imag(rhs));
}
P operator-(const P& p){
    return P(-real(p), -imag(p));
}

ll cross(const P& lhs, const P& rhs){
    return real(lhs)*imag(rhs) - imag(lhs)*real(rhs);
}

ll dot(const P& lhs, const P& rhs){
    return real(lhs)*real(rhs) + imag(lhs)*imag(rhs);
}

ll distance(const P& a, const P& b, const P& p){
    ll c = cross(p-a, b-a);
    return c * c;
}

P ps[3], cc;
ll radius;

bool doesLineAcrossCircle(const P& a, const P& b){
    // if(cross(a-cc, b-cc) == 0){
    //     return dot(a-cc, b-cc) < 0 && norm(a) >= radius * radius && norm(b) >= radius * radius;
    // }
    if(distance(a, b, cc) > radius * radius * norm(b-a)){return false;}
    if(dot(cc-a, b-a) * dot(cc-b, b-a) <= 0){return true;}
    return norm(cc-a) <= radius * radius || norm(cc-b) <= radius * radius;
}

bool isCircleInTriangle(){
    REP(i, 3){
        if(distance(ps[i], ps[(i+1)%3], cc) < radius * radius * norm(ps[i]-ps[(i+1)%3])){return false;}
    }

    int count_n = 0, count_p = 0;
    REP(i, 3){
        ll c = cross(cc-ps[i], ps[(i+1)%3]-ps[i]);
        if(c < 0){
            count_n += 1;
        }else if(c == 0){
            count_n += 1; count_p += 1;
        }else{
            count_p += 1;
        }
    }

    if(count_n % 3 == 0 || count_p % 3 == 0){return true;}
    return false;
}

bool isTriangleInCircle(){
    REP(i, 3){
        if(norm(ps[i]-cc) > radius * radius){return false;}
    }
    return true;
}

bool isIntersected(){
    REP(i, 3){
        if(doesLineAcrossCircle(ps[i], ps[(i+1)%3])){return true;}
    }
    return false;
}

int main(){
    ll re, im;
    while(std::cin >> re >> im, re || im){
        ps[0].real(re); ps[0].imag(im);
        
        FOR(i, 1, 3){
            ll re, im;
            std::cin >> re >> im;
        
            ps[i].real(re); ps[i].imag(im);
        }
        
        {
            ll re, im;
            std::cin >> re >> im;
            
            cc.real(re); cc.imag(im);
        }
        
        std::cin >> radius;
        
        if(isCircleInTriangle()){puts("a");}
        else if(isTriangleInCircle()){puts("b");}
        else if(isIntersected()){puts("c");}
        else{puts("d");}
    }   
}