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<cstring>
#include<algorithm>
#include<math.h>

using namespace std;

class Point {
public:
	double x, y;

	Point() {
		x = y = 0;
	}
};

Point Circle, Triangle[3];
double Radius, ShortestDist[3], DistanceOfPointToPoint[3];
bool isCirclePointInsideTriangleFlag;


bool isCirclePointInsideTriangle() {
	Point vecMA, vecMB, vecMC;
	double a, b, c;

	vecMA.x = Circle.x - Triangle[0].x;
	vecMA.y = Circle.y - Triangle[0].y;
	vecMB.x = Circle.x - Triangle[1].x;
	vecMB.y = Circle.y - Triangle[1].y;
	vecMC.x = Circle.x - Triangle[2].x;
	vecMC.y = Circle.y - Triangle[2].y;

	a = vecMA.x * vecMB.y - vecMA.y * vecMB.x;
	b = vecMB.x * vecMC.y - vecMB.y * vecMC.x;
	c = vecMC.x * vecMA.y - vecMC.y * vecMA.x;

	if ((a <= 0 && b <= 0 && c <= 0) || (a >= 0 && b >= 0 && c >= 0)) {
		return true;
	}	
	return false;
}

//distanceOfp1Top2
double calculationDistanceByPointToPoint(const Point& p1, const Point&p2) {
	return sqrt((p1.x - p2.x)*(p1.x - p2.x) + (p1.y - p2.y)*(p1.y - p2.y));
}

bool bothLess90Degree(const Point &a, const Point &b, const Point &c)
{
	return ((a.x - b.x)*(c.x - b.x) + (a.y - b.y)*(c.y - b.y)) > 0;
}

//distanceOfp1ToLine(p2,p3)
double calculationDistanceByPointToLine(const Point& p1, const Point&p2, const Point&p3) {
	double k, b;

	if (bothLess90Degree(p1, p2, p3) && bothLess90Degree(p1, p3, p2)) {
		if (p2.x == p3.x) {
			return fabs(p1.x - p2.x);
		}

		k = (p3.y - p2.y) / (p3.x - p2.x);
		b = p2.y - k * p2.x;

		return ((fabs(k*p1.x - p1.y + b)) / sqrt(k*k + 1));
	}

	return min(calculationDistanceByPointToPoint(p1, p2), calculationDistanceByPointToPoint(p1, p3));
}

bool isB() {
	for (int i = 0; i < 3; i++) {
		if (DistanceOfPointToPoint[i] > Radius) {
			return false;
		}
	}
	return true;
}

bool isA() {
	for (int i = 0; i < 3; i++) {
		if (ShortestDist[i] < Radius) {
			return false;
		}
	}

	if (isCirclePointInsideTriangleFlag) {
		return true;
	}
	return false;
}

bool isD() {
	for (int i = 0; i < 3; i++) {
		if (ShortestDist[i] < Radius) {
			return false;
		}
		if (fabs(ShortestDist[i] - Radius) < 1e-10) {
			return false;
		}
	}

	if (isCirclePointInsideTriangleFlag) {
		return false;
	}
	return true;
}

void solve() {

	//distanceOfCirclePointToTrianglePoint
	for (int i = 0; i < 3; i++) {
		DistanceOfPointToPoint[i] = calculationDistanceByPointToPoint(Circle,Triangle[i]);
	}
	if ( isB() ) {
		cout << "b" << endl;
		return;
	}


	for (int i = 0; i < 3; i++) {
		ShortestDist[i] = calculationDistanceByPointToLine(Circle, Triangle[i], Triangle[(i+1)%3]);
	}
	if( isA() ){
		cout << "a" << endl;
		return;
	}
	if (isD()) {
		cout << "d" << endl;
		return;
	}

	cout << "c" << endl;
}


int main() {

	while (true) {
		cin >> Triangle[0].x >> Triangle[0].y;
		if (Triangle[0].x == 0 && Triangle[0].y == 0) {
			break;
		}
		for (int i = 1; i < 3; i++) {
			cin >> Triangle[i].x >> Triangle[i].y;
		}
		cin >> Circle.x >> Circle.y >> Radius;

		isCirclePointInsideTriangleFlag = isCirclePointInsideTriangle();
		solve();
	}

	return 0;
}