#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;
}

