Many mathematical methods
First, find the circumcenter; the center of the circumcircle of a regular N-gon is exactly the circumcenter.
After finding it, use the rotation vector to rotate.
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// Author : GCA ||
// 6AE7EE02212D47DAD26C32C0FE829006 ||
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#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
#include <climits>
#include <vector>
#include <set>
#include <map>
#include <queue>
#include <cctype>
#include <utility>
using namespace std;
#ifdef ONLINE\_JUDGE
#define ll "%lld"
#else
#define ll "%I64d"
#endif
typedef unsigned int uint;
typedef long long int Int;
#define Set(a,s) memset(a,s,sizeof(a))
#define Write(w) freopen(w,"w",stdout)
#define Read(r) freopen(r,"r",stdin)
#define Pln() printf("\\n")
#define I\_de(x,n)for(int i=0;i<n;i++)printf("%d ",x\[i\]);Pln()
#define De(x)printf(#x"%d\\n",x)
#define For(i,x)for(int i=0;i<x;i++)
#define CON(x,y) x##y
#define Pmz(dp,nx,ny)for(int hty=0;hty<ny;hty++){for(int htx=0;htx<nx;htx++){\\
printf("%d ",dp\[htx\]\[hty\]);}Pln();}
#define M 55
#define PII pair<int,int\>
#define PB push\_back
#define oo INT\_MAX
#define Set\_oo 0x3f
#define Is\_debug true
#define debug(...) if(Is\_debug)printf("DEBUG: "),printf(\_\_VA\_ARGS\_\_)
#define FOR(it,c) for(\_\_typeof((c).begin()) it=(c).begin();it!=(c).end();it++)
#define eps 1e-6
bool xdy(double x,double y){return x>y+eps;}
bool xddy(double x,double y){return x>y-eps;}
bool xcy(double x,double y){return x<y-eps;}
bool xcdy(double x,double y){return x<y+eps;}
int min3(int x,int y,int z){
int tmp=min(x,y);
return min(tmp,z);
}
int max3(int x,int y,int z){
int tmp=max(x,y);
return max(tmp,z);
}
int n;
int ff;
double pi=acos(-1);
struct point{
double x,y;
}pt\[4\];
inline double cosd(double a){
return cos(a\*pi/180);
}
inline double sind(double a){
return sin(a\*pi/180);
}
point cm(point a,point b,point c){
point ua,ub,va,vb,ans;
ua.x=(a.x+b.x)/2;
ua.y=(a.y+b.y)/2;
ub.x=ua.x-a.y+b.y;
ub.y=ua.y+a.x-b.x;
va.x=(a.x+c.x)/2;
va.y=(a.y+c.y)/2;
vb.x=va.x-a.y+c.y;
vb.y=va.y+a.x-c.x;
// printf("%.3f %.3f %.3f %.3f\\n%.3f %.3f %.3f %.3f\\n",ua.x,ua.y,ub.x,ub.y,va.x,va.y,vb.x,vb.y);
double tmp=((vb.x-ua.x)\*(va.y-vb.y)-(vb.y-ua.y)\*(va.x-vb.x))/
((ub.x-ua.x)\*(va.y-vb.y)-(ub.y-ua.y)\*(va.x-vb.x));
// printf("%.5f %.4f",tmp,ub.y-ua.y);
ans.x=ua.x+(ub.x-ua.x)\*tmp;
ans.y=ua.y+(ub.y-ua.y)\*tmp;
return ans;
}
point wr(point a,point b,double ang){
point ans=a;
ans.x+=(b.x-a.x)\*cos(ang)-(b.y-a.y)\*sin(ang);
ans.y+=(b.x-a.x)\*sin(ang)+(b.y-a.y)\*cos(ang);
return ans;
}
void solve(){
point c=cm(pt\[0\],pt\[1\],pt\[2\]);
// printf("%.2f %.2f\\n",c.x,c.y);
double mxx=-oo,mnx=oo,mxy=-oo,mny=oo;
double ang=2\*pi/n;
for(int i=0;i<n;i++){
point p=wr(c,pt\[0\],ang\*i);
mxx=max(mxx,p.x);
mxy=max(mxy,p.y);
mnx=min(mnx,p.x);
mny=min(mny,p.y);
}
printf("Polygon %d: %.3f\\n",++ff,(mxx-mnx)\*(mxy-mny));
}
int main() {
ios\_base::sync\_with\_stdio(0);
ff=0;
while(~scanf("%d",&n)&&n){
for(int i=0;i<3;i++){
scanf("%lf%lf",&pt\[i\].x,&pt\[i\].y);
}
solve();
}
}