如何判断三角形的外心、重心、内心、垂心？

Question

如题所述

Answer 1

判断三角形的外心、重心、内心、垂心需要根据这些点的定义和求解公式进行判断。下面是每个点的定义和求解公式：

1. 外心：三角形外接圆的圆心，是三条中线的交点。

求解公式：三角形任意两边的垂直平分线的交点即为外心。

2. 重心：三角形三条中线交于一点，这个点叫做重心。

求解公式：三角形三个顶点连线和所对的重心在一条直线上，重心离两个顶点的距离：

3. 内心：三角形内切圆的圆心，是三条角平分线的交点。

求解公式：三角形任意两边和它们夹角的平分线的交点即为内心。

4. 垂心：三角形三条高交于一点，这个点叫做垂心。

求解公式：对于任意三角形ABC，假设D、E、F分别是AB、BC、CA上的垂足，则垂心H为三条垂线AD、BE、CF交点。

要判断一个三角形的各个特殊点，首先需要有三角形顶点的坐标或边长等必要的条件，然后代入对应的求解公式即可求得各个特殊点。

Answer 2

http://hi.baidu.com/ggggwhw/blog/item/9cbd56a84603cafb1f17a242.html

三角形顶点坐标:

A(x1;
y1);

B(x2;
y2);

C(x3;
y3);

④重心G(x4;y4);

x4=(x1+x2+x3)/3;

y4=(y1+y2+y3)/3;

⑤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));

⑥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));

⑦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);