//整數幾何函數庫
//注意某些情況下整數運算會出界!
#define sign(a) ((a)>0?1:(((a)<0?-1:0)))
struct point{int x,y;};
struct line{point a,b;};

//計算cross product (P1-P0)x(P2-P0)
int xmult(point p1,point p2,point p0){
	return (p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}
int xmult(int x1,int y1,int x2,int y2,int x0,int y0){
	return (x1-x0)*(y2-y0)-(x2-x0)*(y1-y0);
}

//計算dot product (P1-P0).(P2-P0)
int dmult(point p1,point p2,point p0){
	return (p1.x-p0.x)*(p2.x-p0.x)+(p1.y-p0.y)*(p2.y-p0.y);
}
int dmult(int x1,int y1,int x2,int y2,int x0,int y0){
	return (x1-x0)*(x2-x0)+(y1-y0)*(y2-y0);
}

//判三點共線
int dots_inline(point p1,point p2,point p3){
	return !xmult(p1,p2,p3);
}
int dots_inline(int x1,int y1,int x2,int y2,int x3,int y3){
	return !xmult(x1,y1,x2,y2,x3,y3);
}

//判點是否在線段上,包括端點和部分重合
int dot_online_in(point p,line l){
	return !xmult(p,l.a,l.b)&&(l.a.x-p.x)*(l.b.x-p.x)<=0&&(l.a.y-p.y)*(l.b.y-p.y)<=0;
}
int dot_online_in(point p,point l1,point l2){
	return !xmult(p,l1,l2)&&(l1.x-p.x)*(l2.x-p.x)<=0&&(l1.y-p.y)*(l2.y-p.y)<=0;
}
int dot_online_in(int x,int y,int x1,int y1,int x2,int y2){
	return !xmult(x,y,x1,y1,x2,y2)&&(x1-x)*(x2-x)<=0&&(y1-y)*(y2-y)<=0;
}

//判點是否在線段上,不包括端點
int dot_online_ex(point p,line l){
	return dot_online_in(p,l)&&(p.x!=l.a.x||p.y!=l.a.y)&&(p.x!=l.b.x||p.y!=l.b.y);
}
int dot_online_ex(point p,point l1,point l2){
	return dot_online_in(p,l1,l2)&&(p.x!=l1.x||p.y!=l1.y)&&(p.x!=l2.x||p.y!=l2.y);
}
int dot_online_ex(int x,int y,int x1,int y1,int x2,int y2){
	return dot_online_in(x,y,x1,y1,x2,y2)&&(x!=x1||y!=y1)&&(x!=x2||y!=y2);
}

//判兩點在直線同側,點在直線上返回0
int same_side(point p1,point p2,line l){
	return sign(xmult(l.a,p1,l.b))*xmult(l.a,p2,l.b)>0;
}
int same_side(point p1,point p2,point l1,point l2){
	return sign(xmult(l1,p1,l2))*xmult(l1,p2,l2)>0;
}

//判兩點在直線異側,點在直線上返回0
int opposite_side(point p1,point p2,line l){
	return sign(xmult(l.a,p1,l.b))*xmult(l.a,p2,l.b)<0;
}
int opposite_side(point p1,point p2,point l1,point l2){
	return sign(xmult(l1,p1,l2))*xmult(l1,p2,l2)<0;
}

//判兩直線平行
int parallel(line u,line v){
	return (u.a.x-u.b.x)*(v.a.y-v.b.y)==(v.a.x-v.b.x)*(u.a.y-u.b.y);
}
int parallel(point u1,point u2,point v1,point v2){
	return (u1.x-u2.x)*(v1.y-v2.y)==(v1.x-v2.x)*(u1.y-u2.y);
}

//判兩直線垂直
int perpendicular(line u,line v){
	return (u.a.x-u.b.x)*(v.a.x-v.b.x)==-(u.a.y-u.b.y)*(v.a.y-v.b.y);
}
int perpendicular(point u1,point u2,point v1,point v2){
	return (u1.x-u2.x)*(v1.x-v2.x)==-(u1.y-u2.y)*(v1.y-v2.y);
}

//判兩線段相交,包括端點和部分重合
int intersect_in(line u,line v){
	if (!dots_inline(u.a,u.b,v.a)||!dots_inline(u.a,u.b,v.b))
		return !same_side(u.a,u.b,v)&&!same_side(v.a,v.b,u);
	return dot_online_in(u.a,v)||dot_online_in(u.b,v)||dot_online_in(v.a,u)||dot_online_in(v.b,u);
}
int intersect_in(point u1,point u2,point v1,point v2){
	if (!dots_inline(u1,u2,v1)||!dots_inline(u1,u2,v2))
		return !same_side(u1,u2,v1,v2)&&!same_side(v1,v2,u1,u2);
	return dot_online_in(u1,v1,v2)||dot_online_in(u2,v1,v2)||dot_online_in(v1,u1,u2)||dot_online_in(v2,u1,u2);
}


//判兩線段相交,不包括端點和部分重合
int intersect_ex(line u,line v){
	return opposite_side(u.a,u.b,v)&&opposite_side(v.a,v.b,u);
}
int intersect_ex(point u1,point u2,point v1,point v2){
	return opposite_side(u1,u2,v1,v2)&&opposite_side(v1,v2,u1,u2);
}