已知三角形三点坐标A(x1,y1),B(x2,y2),C(x3,y3),如何用这三点坐标表示该三角形的五心？

首先求三边的长
a=√[(x2-x3)²+(y2-y3)²]，b=√[(x1-x3)²+(y1-y3)²]，c=√[(x1-x2)²+(y1-y2)²]
然后设
Ka= -a²+b²+c²，Kb= -b²+a²+c²，Kc= -c²+a²+b²
重心坐标
x重=(x1+x2+x3)/3
y重=(y1+y2+y3)/3
内心坐标
x内=(ax1+bx2+cx3)/(a+b+c)
y内=(ay1+by2+cy3)/(a+b+c)
垂心坐标
x垂=(x1/Ka+x2/Kb+x3/Kc)/(1/Ka+1/Kb+1/Kc)
y垂=(y1/Ka+y2/Kb+y3/Kc)/(1/Ka+1/Kb+1/Kc)
外心坐标
x外=(a²Kax1+b²Kbx2+c²Kcx3)/(a²Ka+b²Kb+c²Kc)
y外=(a²Kay1+b²Kby2+c²Kcy3)/(a²Ka+b²Kb+c²Kc)
旁心坐标
x旁1=(-ax1+bx2+cx3)/(-a+b+c)
y旁1=(-ay1+by2+cy3)/(-a+b+c)
x旁2=(ax1-bx2+cx3)/(a-b+c)
y旁2=(ay1-by2+cy3)/(a-b+c)
x旁3=(ax1+bx2-cx3)/(a+b-c)
y旁3=(ay1+by2-cy3)/(a+b-c)

已知任意三角形ABC三点坐标分别为A(X1,Y1),B(X2,Y2),C(X3,Y3)
求：
1.该三角形重心坐标
2.该三角形内心坐标(三条角平分线交点)
3.该三角形垂心坐标（三条高交点）
4.改三角形外心坐标（三条边垂直平分线交点）
上述四题请简述过程，用含有X1,X2,X3,Y1,Y2,Y3的代数式表示

重心G(x4;y4);
x4=(x1+x2+x3)/3;
y4=(y1+y2+y3)/3;
外心W(x5;y5);
根据外心到各顶点的距离相等:
AG=BG;
AG=CG;
即:
Sqrt[(x1 - x5)^2 + (y1 - y5)^2] == Sqrt[(x2 - x5)^2 + (y2 - y5)^2],
Sqrt[(x1 - x5)^2 + (y1 - y5)^2] == Sqrt[(x3 - x5)^2 + (y3 - y5)^2]
解得:
x5 = (x2^2 y1 - x3^2 y1 - x1^2 y2 + x3^2 y2 - y1^2 y2 + y1 y2^2 + x1^2 y3 - x2^2 y3 + y1^2 y3 - y2^2 y3 - y1 y3^2 + y2 y3^2)/(2 (x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3));
y5 = -(-x1^2 x2 + x1 x2^2 + x1^2 x3 - x2^2 x3 - x1 x3^2 + x2 x3^2 - x2 y1^2 + x3 y1^2 + x1 y2^2 - x3 y2^2 - x1 y3^2 + x2 y3^2)/(2 (x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3));
内心N(x6;y6);
根据内心到各边的距离相等:
先求内心到各边垂线垂足与顶点的距离;
1/2 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] - Sqrt[(x2 - x3)^2 + (y2 - y3)^2]);
1/2 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] - Sqrt[(x1 - x3)^2 + (y1 - y3)^2] + Sqrt[(x2 - x3)^2 + (y2 - y3)^2]);
1/2 (-Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] + Sqrt[(x2 - x3)^2 + (y2 - y3)^2]);
计算内心到个顶点的距离;根据勾股定理计算内心到各边的距离,根据距离相等列方程:
(x1 - x6)^2 - 1/4 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] - Sqrt[(x2 - x3)^2 + (y2 - y3)^2])^2 + (y1 - y6)^2 == (x2 - x6)^2 - 1/4 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] - Sqrt[(x1 - x3)^2 + (y1 - y3)^2] + Sqrt[(x2 - x3)^2 + (y2 - y3)^2])^2 + (y2 - y6)^2,
(x1 - x6)^2 - 1/4 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] - Sqrt[(x2 - x3)^2 + (y2 - y3)^2])^2 + (y1 - y6)^2 == (x3 - x6)^2 - 1/4 (-Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] + Sqrt[(x2 - x3)^2 + (y2 - y3)^2])^2 + (y3 - y6)^2
解得:
x6 = (x2^2 y1 - x3^2 y1 - x1^2 y2 + x3^2 y2 - y1^2 y2 + y1 y2^2 + x1^2 y3 - x2^2 y3 + y1^2 y3 - y2^2 y3 - y1 y3^2 + y2 y3^2 + y2 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] - Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] y3 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] - y1 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] + Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] y3 Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] + y1 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] - y2 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2])/(2 (x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3));
y6 = -(-x1^2 x2 + x1 x2^2 + x1^2 x3 - x2^2 x3 - x1 x3^2 + x2 x3^2 - x2 y1^2 + x3 y1^2 + x1 y2^2 - x3 y2^2 - x1 y3^2 + x2 y3^2 + x2 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] - x3 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] - x1 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] + x3 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] + x1 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] - x2 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2])/(2 (x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3));
垂心H(x7;y7);
分别做高线: AH⊥BC;BH⊥AC;
(y1 - y7)/(x1 - x7) (y2 - y3)/(x2 - x3) == -1,
(y2 - y7)/(x2 - x7) (y1 - y3)/(x1 - x3) == -1
解得:
x7 = -(x1 x2 y1 - x1 x3 y1 - x1 x2 y2 + x2 x3 y2 + y1^2 y2 - y1 y2^2 + x1 x3 y3 - x2 x3 y3 - y1^2 y3 + y2^2 y3 + y1 y3^2 - y2 y3^2)/(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3);
y7 = -(x1^2 x2 - x1 x2^2 - x1^2 x3 + x2^2 x3 + x1 x3^2 - x2 x3^2 + x1 y1 y2 - x2 y1 y2 - x1 y1 y3 + x3 y1 y3 + x2 y2 y3 - x3 y2 y3)/(x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3);

采纳我吧，哥哥