計算三角形費馬點

1.在一個三角形中，到3個頂點距離之和最小的點叫做這個三角形的費馬點。

2.　費馬點計算方法:

(1)若三角形ABC的3個內角均小於120°，那麼3條距離連線正好平分費馬點所在的周角。所以三角形的費馬點也稱爲三角形的等角中心。

(2)若三角形有一內角不小於120度，則此鈍角的頂點就是距離和最小的點。

3.如何計算等角中心呢？

做任意一條邊的外接等邊三角形，得到另一點，將此點與此邊在三角形中對應的點相連

如此再取另一邊作同樣的連線，相交點即費馬點

證明畫幾條輔助線就出來了~這裏就不證明了

這是合肥OI競賽一次比賽的題目，去給50中小朋友上課時寫的標程:

#include <iostream>
#include <math.h>
#include <windows.h>
#include <iomanip>
using namespace std;

struct Vec
{
	double x,y;
	Vec(double xx=0,double yy=0)
	{
		x=xx;
		y=yy;
	}
};

struct Point
{
	double x,y;
	Point(double xx=0,double yy=0)
	{
		x=xx;
		y=yy;
	}
};

double ddot(Vec A,Vec B)
{
	return A.x*B.x+A.y*B.y;
}
double getlen(Vec A)
{
	return sqrt(A.x*A.x+A.y*A.y);
}

double getlen(Point A,Point B)
{
	return sqrt((A.x-B.x)*(A.x-B.x)+(A.y-B.y)*(A.y-B.y));
}

bool moreThan120(double xa,double ya,double xb,double yb,double xc,double yc)
{
	Vec ab(xb-xa,yb-ya),ac(xc-xa,yc-ya);
	if (ddot(ab,ac)/getlen(ab)/getlen(ac) < -0.5)
	{
		return true;
	}
	return false;
}

inline void swap(double &a,double &b)
{
	double t;
	t=a;
	a=b;
	b=t;
}


Point getAnotherPoint(Point A,Point B,Point C)
{
	Point r,C1,C2;
	Vec AB(B.x-A.x,B.y-A.y);
	double len,len2;
	double sqrt3=sqrt(3.0);
	Vec AB2,crossAB,crossAB2;

	AB2.x=AB.x/2; AB2.y=AB.y/2;
	crossAB.x=AB2.y; crossAB.y=-AB2.x;
	crossAB2.x=-AB2.y; crossAB2.y=AB2.x;

	len=getlen(AB2);
	
	crossAB.x*=sqrt3; crossAB.y*=sqrt3;
	crossAB2.x*=sqrt3; crossAB2.y*=sqrt3;

	C1.x=A.x+AB2.x+crossAB.x;C1.y=A.y+AB2.y+crossAB.y;
	C2.x=A.x+AB2.x+crossAB2.x;C2.y=A.y+AB2.y+crossAB2.y;

	if (getlen(C,C1)<getlen(C,C2))
	{
		return C2;
	}else
		return C1;

}


/*
在平面內的兩條直線AB,CD，求交點最直接的方法就是解下列的二元二次方程組：
Ax + (Bx - Ax)i = Cx + (Dx - Cx) j
Ay + (By - Ay)i = Cy + (Dy - Cy) j
交點是:
(Ax + (Bx - Ax) i, Ay + (By - Ay) i)
即：
  Ax + (AAx)i = Bx + (BBx) j
  Ay + (AAy)i = By + (BBy) j
  交點是:
(Ax + (AAx)i, Ay + (AAy)i)
*/
Point getCrossPoint(Point A,Point A1,Point B,Point B1)
{
	Point r;
	Vec AA(A1.x-A.x,A1.y-A.y),BB(B1.x-B.x,B1.y-B.y);
	double i,j,tmp,tmp2;
	double Ax=A.x,Ay=A.y,AAx=AA.x,AAy=AA.y,Bx=B.x,By=B.y,BBx=BB.x,BBy=BB.y;

	if (AAx==0)
	{
		j=(Ax-Bx)/BBx;
		i=(By+BBy*j-Ay)/AAx;
	}else if (BBx==0)
	{
		i=(Bx-Ax)/AAx;
	}else if (AAy==0)
	{
		j=(Ay-By)/BBy;
		i=(Bx-Ax-BBx*j)/AAx;
	}else if (BBy==0)
	{
		i=(By-Ay)/AAy;
	}
	else
	{
		tmp=AAx;
		tmp2=AAy;
		Ax*=AAy;AAx*=AAy;Bx*=AAy;BBx*=AAy;
		Ay*=tmp;AAy*=tmp;By*=tmp;BBy*=tmp;
		j=((Ax-Ay)-(Bx-By))/(BBx-BBy);
		i=(Bx+BBx*j-Ax)/AAx;
	}

	r.x=(Ax+AAx*i)/tmp2;
	r.y=(Ay+AAy*i)/tmp;
	return r;
}

int main()
{
	freopen("cul.in9","r",stdin);

	double xa,ya,xb,yb,xc,yc;
	Point C1,A1,R;


	cin>>xa>>ya>>xb>>yb>>xc>>yc;
	cout.setf(ios::fixed);
	
	if (moreThan120(xa,ya,xb,yb,xc,yc))
	{
		cout<<setprecision(10)<<xa<<" "<<ya<<endl; 
	}else if (moreThan120(xb,yb,xa,ya,xc,yc))
	{
		cout<<setprecision(10)<<xb<<" "<<yb<<endl; 
	}else if (moreThan120(xc,yc,xa,ya,xb,yb))
	{
		cout<<setprecision(10)<<xc<<" "<<yc<<endl; 
	}else
	{
		C1=getAnotherPoint(Point(xa,ya),Point(xb,yb),Point(xc,yc));
		A1=getAnotherPoint(Point(xc,yc),Point(xb,yb),Point(xa,ya));

		R=getCrossPoint(Point(xa,ya),A1,Point(xc,yc),C1);
		cout<<setprecision(10)<<R.x<<" "<<R.y<<endl;
	}
	Sleep(1000000);
}