uva 10577

很多數學方法

先求出外心，正N邊形的外接圓的中心剛好是外心

求出之後再用旋轉向量去繞

//====================================================================||
//                                                                    ||
//                                                                    ||
//                         Author : GCA                               ||
//                  6AE7EE02212D47DAD26C32C0FE829006                  ||
//====================================================================||
#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();
    }

















}