#include <math.h>
struct point{double x,y;};
struct line{point a,b;};

double distance(point p1,point p2){
	return sqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y));
}

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;
}

//外心
point circumcenter(point a,point b,point c){
	line u,v;
	u.a.x=(a.x+b.x)/2;
	u.a.y=(a.y+b.y)/2;
	u.b.x=u.a.x-a.y+b.y;
	u.b.y=u.a.y+a.x-b.x;
	v.a.x=(a.x+c.x)/2;
	v.a.y=(a.y+c.y)/2;
	v.b.x=v.a.x-a.y+c.y;
	v.b.y=v.a.y+a.x-c.x;
	return intersection(u,v);
}

//內(nèi)心
point incenter(point a,point b,point c){
	line u,v;
	double m,n;
	u.a=a;
	m=atan2(b.y-a.y,b.x-a.x);
	n=atan2(c.y-a.y,c.x-a.x);
	u.b.x=u.a.x+cos((m+n)/2);
	u.b.y=u.a.y+sin((m+n)/2);
	v.a=b;
	m=atan2(a.y-b.y,a.x-b.x);
	n=atan2(c.y-b.y,c.x-b.x);
	v.b.x=v.a.x+cos((m+n)/2);
	v.b.y=v.a.y+sin((m+n)/2);
	return intersection(u,v);
}

//垂心
point perpencenter(point a,point b,point c){
	line u,v;
	u.a=c;
	u.b.x=u.a.x-a.y+b.y;
	u.b.y=u.a.y+a.x-b.x;
	v.a=b;
	v.b.x=v.a.x-a.y+c.y;
	v.b.y=v.a.y+a.x-c.x;
	return intersection(u,v);
}

//重心
//到三角形三頂點(diǎn)距離的平方和最小的點(diǎn)
//三角形內(nèi)到三邊距離之積最大的點(diǎn)
point barycenter(point a,point b,point c){
	line u,v;
	u.a.x=(a.x+b.x)/2;
	u.a.y=(a.y+b.y)/2;
	u.b=c;
	v.a.x=(a.x+c.x)/2;
	v.a.y=(a.y+c.y)/2;
	v.b=b;
	return intersection(u,v);
}

//費(fèi)馬點(diǎn)
//到三角形三頂點(diǎn)距離之和最小的點(diǎn)
point fermentpoint(point a,point b,point c){
	point u,v;
	double step=fabs(a.x)+fabs(a.y)+fabs(b.x)+fabs(b.y)+fabs(c.x)+fabs(c.y);
	int i,j,k;
	u.x=(a.x+b.x+c.x)/3;
	u.y=(a.y+b.y+c.y)/3;
	while (step>1e-10)
		for (k=0;k<10;step/=2,k++)
			for (i=-1;i<=1;i++)
				for (j=-1;j<=1;j++){
					v.x=u.x+step*i;
					v.y=u.y+step*j;
					if (distance(u,a)+distance(u,b)+distance(u,c)>distance(v,a)+distance(v,b)+distance(v,c))
						u=v;
				}
	return u;
}