数学一本通——K. 点与三角形关系（叉积求面积）

double cul(node a,node b,node c){
    double x1=a.x,y1=a.y;
    double x2=b.x,y2=b.y;
    double x3=c.x,y3=c.y;
    return x1*y2-x1*y3+x2*y3-x2*y1+x3*y1-x2*y2;
}

double cul(node a,node b,node c)
{
    double x1=a.x,y1=a.y;
    double x2=b.x,y2=b.y;
    double x3=c.x,y3=c.y;
    double aa=sqrt( (x1-x2)*(x1-x2) + (y1-y2)*(y1-y2) );
    double bb=sqrt( (x3-x2)*(x3-x2) + (y3-y2)*(y3-y2) );
    double cc=sqrt( (x1-x3)*(x1-x3) + (y1-y3)*(y1-y3) );
    double pp=(aa+bb+cc)/2;
    double S=sqrt( pp*(pp-aa)*(pp-bb)*(pp-cc));
    return S;
}

#pragma GCC optimize(2)
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
typedef pair<ll,ll>PLL;
typedef pair<int,int>PII;
typedef pair<double,double>PDD;
#define I_int ll
inline ll read()
{
    ll x=0,f=1;
    char ch=getchar();
    while(ch<'0'||ch>'9')
    {
        if(ch=='-')f=-1;
        ch=getchar();
    }
    while(ch>='0'&&ch<='9')
    {
        x=x*10+ch-'0';
        ch=getchar();
    }
    return x*f;
}
#define read read()
#define closeSync ios::sync_with_stdio(0);cin.tie(0);cout.tie(0)
#define multiCase int T;cin>>T;for(int t=1;t<=T;t++)
#define rep(i,a,b) for(int i=(a);i<=(b);i++)
#define repp(i,a,b) for(int i=(a);i<(b);i++)
#define per(i,a,b) for(int i=(a);i>=(b);i--)
#define perr(i,a,b) for(int i=(a);i>(b);i--)
ll ksm(ll a,ll b,ll p)
{
    ll res=1;
    while(b)
    {
        if(b&1)res=res*a%p;
        a=a*a%p;
        b>>=1;
    }
    return res;
}
#define PI acos(-1)
const double eps=1e-10;
struct node
{
    double x,y;
    node operator - (node &s){
        return (node){x-s.x,y-s.y};
    }
};
double operator*(node a,node b){
    return a.x*b.y-b.x*a.y;
}
double cul(node a,node b,node c){
    return fabs((b-a)*(c-a)/2);
}
int main()
{
    node a,b,c,d;
    cin>>a.x>>a.y>>b.x>>b.y>>c.x>>c.y>>d.x>>d.y;
    double abc=cul(a,b,c);
    double abd=cul(a,b,d);
    double acd=cul(a,c,d);
    double bcd=cul(b,c,d);
    if(abd+acd+bcd==abc) puts("in");
    else puts("out");
    return 0;
}