//浮點幾何函數庫
#include <math.h>
#define eps 1e-8
#define zero(x) (((x)>0?(x):-(x))<eps)
struct point{double x,y;};
struct line{point a,b;};

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

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

//兩點距離
double distance(point p1,point p2){
	return sqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y));
}
double distance(double x1,double y1,double x2,double y2){
	return sqrt((x1-x2)*(x1-x2)+(y1-y2)*(y1-y2));
}

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

//判點是否在線段上,包括端點
int dot_online_in(point p,line l){
	return zero(xmult(p,l.a,l.b))&&(l.a.x-p.x)*(l.b.x-p.x)<eps&&(l.a.y-p.y)*(l.b.y-p.y)<eps;
}
int dot_online_in(point p,point l1,point l2){
	return zero(xmult(p,l1,l2))&&(l1.x-p.x)*(l2.x-p.x)<eps&&(l1.y-p.y)*(l2.y-p.y)<eps;
}
int dot_online_in(double x,double y,double x1,double y1,double x2,double y2){
	return zero(xmult(x,y,x1,y1,x2,y2))&&(x1-x)*(x2-x)<eps&&(y1-y)*(y2-y)<eps;
}

//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)&&(!zero(p.x-l.a.x)||!zero(p.y-l.a.y))&&(!zero(p.x-l.b.x)||!zero(p.y-l.b.y));
}
int dot_online_ex(point p,point l1,point l2){
	return dot_online_in(p,l1,l2)&&(!zero(p.x-l1.x)||!zero(p.y-l1.y))&&(!zero(p.x-l2.x)||!zero(p.y-l2.y));
}
int dot_online_ex(double x,double y,double x1,double y1,double x2,double y2){
	return dot_online_in(x,y,x1,y1,x2,y2)&&(!zero(x-x1)||!zero(y-y1))&&(!zero(x-x2)||!zero(y-y2));
}

//判兩點在線段同側,點在線段上返回0
int same_side(point p1,point p2,line l){
	return xmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)>eps;
}
int same_side(point p1,point p2,point l1,point l2){
	return xmult(l1,p1,l2)*xmult(l1,p2,l2)>eps;
}

//判兩點在線段異側,點在線段上返回0
int opposite_side(point p1,point p2,line l){
	return xmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)<-eps;
}
int opposite_side(point p1,point p2,point l1,point l2){
	return xmult(l1,p1,l2)*xmult(l1,p2,l2)<-eps;
}

// 點關于直線的對稱點 // by lyt
// 缺點：用了斜率
// 也可以利用"點到直線上的最近點"來做，避免使用斜率。
point symmetric_point(point p1, point l1, point l2) {
  point ret;  
  if (l1.x > l2.x - eps && l1.x < l2.x + eps) {
    ret.x = (2 * l1.x - p1.x);
	ret.y = p1.y;
  } else {
    double k = (l1.y - l2.y ) / (l1.x - l2.x);
    ret.x = (2*k*k*l1.x + 2*k*p1.y - 2*k*l1.y - k*k*p1.x + p1.x) / (1 + k*k);
    ret.y = p1.y - (ret.x - p1.x ) / k;
  }
  return ret;
}

//判兩直線平行
int parallel(line u,line v){
	return zero((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 zero((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 zero((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 zero((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);
}

//計算兩直線交點,注意事先判斷直線是否平行!
//線段交點請另外判線段相交(同時還是要判斷是否平行!)
point intersection(line u,line v){
	point ret=u.a;
	double t=((u.a.x-v.a.x)*(v.a.y-v.b.y)-(u.a.y-v.a.y)*(v.a.x-v.b.x))
			/((u.a.x-u.b.x)*(v.a.y-v.b.y)-(u.a.y-u.b.y)*(v.a.x-v.b.x));
	ret.x+=(u.b.x-u.a.x)*t;
	ret.y+=(u.b.y-u.a.y)*t;
	return ret;
}
point intersection(point u1,point u2,point v1,point v2){
	point ret=u1;
	double t=((u1.x-v1.x)*(v1.y-v2.y)-(u1.y-v1.y)*(v1.x-v2.x))
			/((u1.x-u2.x)*(v1.y-v2.y)-(u1.y-u2.y)*(v1.x-v2.x));
	ret.x+=(u2.x-u1.x)*t;
	ret.y+=(u2.y-u1.y)*t;
	return ret;
}

//點到直線上的最近點
point ptoline(point p,line l){
	point t=p;
	t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
	return intersection(p,t,l.a,l.b);
}
point ptoline(point p,point l1,point l2){
	point t=p;
	t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;
	return intersection(p,t,l1,l2);
}

//點到直線距離
double disptoline(point p,line l){
	return fabs(xmult(p,l.a,l.b))/distance(l.a,l.b);
}
double disptoline(point p,point l1,point l2){
	return fabs(xmult(p,l1,l2))/distance(l1,l2);
}
double disptoline(double x,double y,double x1,double y1,double x2,double y2){
	return fabs(xmult(x,y,x1,y1,x2,y2))/distance(x1,y1,x2,y2);
}

//點到線段上的最近點
point ptoseg(point p,line l){
	point t=p;
	t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
	if (xmult(l.a,t,p)*xmult(l.b,t,p)>eps)
		return distance(p,l.a)<distance(p,l.b)?l.a:l.b;
	return intersection(p,t,l.a,l.b);
}
point ptoseg(point p,point l1,point l2){
	point t=p;
	t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;
	if (xmult(l1,t,p)*xmult(l2,t,p)>eps)
		return distance(p,l1)<distance(p,l2)?l1:l2;
	return intersection(p,t,l1,l2);
}

//點到線段距離
double disptoseg(point p,line l){
	point t=p;
	t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
	if (xmult(l.a,t,p)*xmult(l.b,t,p)>eps)
		return distance(p,l.a)<distance(p,l.b)?distance(p,l.a):distance(p,l.b);
	return fabs(xmult(p,l.a,l.b))/distance(l.a,l.b);
}
double disptoseg(point p,point l1,point l2){
	point t=p;
	t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;
	if (xmult(l1,t,p)*xmult(l2,t,p)>eps)
		return distance(p,l1)<distance(p,l2)?distance(p,l1):distance(p,l2);
	return fabs(xmult(p,l1,l2))/distance(l1,l2);
}

//矢量V以P為頂點逆時針旋轉angle并放大scale倍
point rotate(point v,point p,double angle,double scale){
	point ret=p;
	v.x-=p.x,v.y-=p.y;
	p.x=scale*cos(angle);
	p.y=scale*sin(angle);
	ret.x+=v.x*p.x-v.y*p.y;
	ret.y+=v.x*p.y+v.y*p.x;
	return ret;
}