坐标原点\(o\),三角形外心\(O\)，重心\(G\)，垂心\(H\)，内心\(I\)
来自老早以前的纸笔记本上，不保证正确性
upd 2021-02-01 验证正确性的话，上mma，找到TriangleCenter函数，然后自己算一下，比较一下。或者你看这个
三角形重心\(G\)
向量
\[\vec{oG}=\frac{1}{3}(\vec{oA}+\vec{oB}+\vec{oC}) \]直角坐标
\[x_G=\frac{1}{3}({x_1+x_2+x_3}) \] \[y_G=\frac{1}{3}({y_1+y_2+y_3}) \]三角形外心\(O\)
向量
\[sin(2A)\cdot\vec{OA}+sin(2B)\cdot\vec{OB}++sin(2C)\cdot\vec{OC}=\vec{0} \] \[\vec{oO}=\frac{sin(2A)\cdot\vec{oA}+sin(2B)\cdot\vec{oB}++sin(2C)\cdot\vec{oC}}{sin(2A)+sin(2B)+sin(2C)} \]直角坐标
\[\begin{aligned} x_O& =\frac{ \left|\begin{array}{cccc} \frac{x_2^2-x_3^2}{2}+\frac{y_2^2-y_3^2}{2} & y_2-y_3 \\ \frac{x_3^2-x_1^2}{2}+\frac{y_3^2-y_1^2}{2} & y_3-y_1 \\ \end{array}\right| }{ \left|\begin{array}{cccc} x_2-x_3 & y_2-y_3 \\ x_3-x_1 & y_3-y_1 \\ \end{array}\right| } \\ &=\frac{[x_1^2(y_2-y_3)+x_2^2(y_3-y_1)+x_3^2(y_1-y_2)]-(y_1-y_2)(y_2-y_3)(y_3-y_1)} {2[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]} \end{aligned} \] \[\begin{aligned} y_O&=\frac{ \left|\begin{array}{cccc} x_2-x_3 & \frac{x_2^2-x_3^2}{2}+\frac{y_2^2-y_3^2}{2} \\ x_3-x_1 & \frac{x_3^2-x_1^2}{2}+\frac{y_3^2-y_1^2}{2} \\ \end{array}\right| }{ \left|\begin{array}{cccc} x_2-x_3 & y_2-y_3 \\ x_3-x_1 & y_3-y_1 \\ \end{array}\right| } \\ &=\frac{-[y_1^2(x_2-x_3)+y_2^2(x_3-x_1)+y_3^2(x_1-x_2)]+(x_1-x_2)(x_2-x_3)(x_3-x_1)} {2[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]} \end{aligned} \]三角形垂心\(H\)
向量
\[\vec{0}=tan(A)\cdot \vec{HA}+tan(B)\cdot \vec{HB}+tan(C)\cdot \vec{HC} \] \[\vec{oH}=\frac{tan(A)\cdot \vec{oA}+tan(B)\cdot \vec{oB}+tan(C)\cdot \vec{oC}}{tan(A)+tan(B)+tan(C)} \]直角坐标
\[x_H=\frac{(y_1-y_2)(y_2-y_3)(y_3-y_1)-[x_1x_2(y_1-y_2)+x_2x_3(y_2-y_3)+x_3x_1(y_3-y_1)]} {x_3(y_1-y_2)+x_1(y_2-y_3)+x_2(y_3-y_1)} \] \[y_H=\frac{-(x_1-x_2)(x_2-x_3)(x_3-x_1)+[y_1y_2(x_1-x_2)+y_2y_3(x_2-x_3)+y_3y_1(x_3-x_1)]} {x_3(y_1-y_2)+x_1(y_2-y_3)+x_2(y_3-y_1)} \]三角形内心\(I\)
向量
\[\vec{0}=a\cdot \vec{IA}+b\cdot \vec{IB}+c\cdot \vec{IC} \] \[\vec{oI}=\frac{a\cdot \vec{oA}+b\cdot \vec{oB}+c\cdot \vec{oC}}{a+b+c} \]直角坐标
\[x_I=\frac{ax_A+bx_B+cx_C}{a+b+c} \] \[y_I=\frac{ay_A+by_B+cy_C}{a+b+c} \]编辑公式不易，转载请注明出处