Triangular garden

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题意：判断一个点是否为三角形四心（重心、垂心、内心、外心）的任意一个。

分析：写这篇博客主要是为了复习几何模板。重心是三条中线的交点坐标为（(x1+x2+x3)/3，(y1+y2+y3)/3）；垂心是三条高线的交点，坐标利用线段垂直的性质（向量的内积为0）解一个方程组；内心是内接圆的圆心，也是利用线段垂直的性质（向量内积为0）解一个方程组；外心是外接圆的圆心，利用到三个点顶点距离相同的性质解一个方程组即可。其实这题没必要求出四心坐标，带进去判断就行了。出题人本来想卡double精度的，但是失败了，本来是只能使用long long的。标程会在后面给出。另外（设向量a（x1，y1）向量b（x2，y2）  a∥b表示a=λb，即x1*y2-x2*y1=0  a⊥b表示a*b=0，即x1x2+y1y2=0）。

我的代码：

#include<bits/stdc++.h>
using namespace std;
#define eps 1e-7
struct p{
    double x,y;
};
///外心
p out(p a,p b,p c){
    p s;
    double a1=2*(b.x-a.x),b1=2*(b.y-a.y),a2=2*(c.x-a.x),b2=2*(c.y-a.y);
    double c1=b.x*b.x-a.x*a.x+b.y*b.y-a.y*a.y,c2=c.x*c.x-a.x*a.x+c.y*c.y-a.y*a.y;
    s.x=(b1*c2-b2*c1)/(a2*b1-a1*b2);
    s.y=(a2*c1-a1*c2)/(a2*b1-a1*b2);
    return s;
}
///内心
p in(p a,p b,p c){
    p s;
    double aa=sqrt((c.y-b.y)*(c.y-b.y)+(c.x-b.x)*(c.x-b.x));
    double bb=sqrt((c.y-a.y)*(c.y-a.y)+(c.x-a.x)*(c.x-a.x));
    double cc=sqrt((b.y-a.y)*(b.y-a.y)+(b.x-a.x)*(b.x-a.x));
    s.x=(aa*a.x+bb*b.x+cc*c.x)/(aa+bb+cc);
    s.y=(aa*a.y+bb*b.y+cc*c.y)/(aa+bb+cc);
    return s;
}
///重心
p zhong(p a, p b,p c){
    p s;
    s.x=(a.x+b.x+c.x)/3,s.y=(a.y+b.y+c.y)/3;
    return s;
}
///垂心
p chui(p a,p b,p c){
    p s;
    double a1=c.x-b.x,b1=c.y-b.y,a2=b.x-a.x,b2=b.y-a.y;
    double c1=a.x*(c.x-b.x)+a.y*(c.y-b.y);
    double c2=c.x*(b.x-a.x)+c.y*(b.y-a.y);
    s.x=(b1*c2-b2*c1)/(a2*b1-a1*b2);
    s.y=(a2*c1-a1*c2)/(a2*b1-a1*b2);
    return s;
}

bool ck(p a,p b){
    if(fabs(a.x-b.x)<eps && fabs(a.y-b.y)<eps ) return true;
    return false;
}

int main(){
    p a,b,c,d;
    while(cin>>a.x>>a.y>>b.x>>b.y>>c.x>>c.y>>d.x>>d.y)
        if( ck(d,zhong(a,b,c)) || ck(d,chui(a,b,c)) || ck(d,out(a,b,c)) || ck(d,in(a,b,c)) ) cout<<"Yes"<<endl;
        else cout<<"No"<<endl;
}

标程：

#include <bits/stdc++.h>
using namespace std;
#define zeros(a,n) memset(a,0,(n)*sizeof(a[0]))
typedef long long LL;
typedef unsigned long long ULL;
const LL modn = 1e9+7;

struct point
{
    LL x, y;
    point operator +(const point &rhs) const
    {
        return (point){x + rhs.x, y + rhs.y};
    }
    point operator -(const point &rhs) const
    {
        return (point){x - rhs.x, y - rhs.y};
    }
    point operator *(const LL &rhs) const
    {
        return (point){x * rhs, y * rhs};
    }
    LL operator *(const point &rhs) const
    {
        return x * rhs.x + y * rhs.y;
    }
    bool operator==(const point &rhs) const
    {
        return x == rhs.x && y == rhs.y;
    }
};

LL quickpow(LL a, LL n, LL p)
{
    LL ans = 1;
    while(n) {
        if(n & 1) ans = ans * a % p;
        a = a * a % p;
        n >>= 1;
    }
    return ans;
}
LL sqr(LL x)
{
    return x * x;
}

LL xmult(const point &lhs, const point &rhs)
{
    return lhs.x * rhs.y - rhs.x * lhs.y;
}

bool isorthocenter(point a, point b, point c, point o)
{
    return (b-a)*(c-o)==0 && (c-b)*(a-o)==0 && (a-c)*(b-o)==0;
}
bool isbarycenter(point a, point b, point c, point o)
{
    return a + b + c == o*3;
}
bool isexcenter(point a, point b, point c, point o)
{
    LL la = (o-a)*(o-a);
    LL lb = (o-b)*(o-b);
    LL lc = (o-c)*(o-c);
    return la == lb && lb == lc;
}
bool isincenter(point a, point b, point c, point o)
{
    LL ta = xmult(b-a, o-a);
    LL tb = xmult(c-b, o-b);
    LL tc = xmult(a-c, o-c);
    if(ta < 0 || tb < 0 || tc < 0) return false;
    LL la = sqr(ta%modn) % modn * quickpow((b-a)*(b-a)%modn, modn-2, modn) % modn;
    LL lb = sqr(tb%modn) % modn * quickpow((c-b)*(c-b)%modn, modn-2, modn) % modn;
    LL lc = sqr(tc%modn) % modn * quickpow((a-c)*(a-c)%modn, modn-2, modn) % modn;
    return la == lb && lb == lc;
}

int main()
{
    point a, b, c, o;
    scanf("%lld %lld", &a.x, &a.y);
    scanf("%lld %lld", &b.x, &b.y);
    scanf("%lld %lld", &c.x, &c.y);
    scanf("%lld %lld", &o.x, &o.y);
    if(xmult(b-a, c-a) < 0) swap(b, c);
    bool flag = false;
    flag |= isorthocenter(a, b, c, o);
    flag |= isbarycenter(a, b, c, o);
    flag |= isexcenter(a, b, c, o);
    flag |= isincenter(a, b, c, o);
    if(flag) puts("Yes");
    else puts("No");
    return 0;
}