问星空：内角平分线三角形？

陈九章

隆重感受到符号运算软件的巨大威力。
好想学哦！

陈九章

利用空间解析几何的知识，容易证明：
四面体的三条中位线交于一点，并且该点平分这三条中位线。
【定义】称四面体的三条中位线的交点为四面体的第二重心。
【注】四面体的第一重心通常定义为顶点与对面三角形重心连线（四条）的交点。
【请问】四面体的第一重心和第二重心，一般情况下应该不会重合，有何关系？

陈九章

hujunhua

四面体的六条棱的中点是一个八面体的顶点，反过来，任意八面体也可以扩展为一个四面体。
如图所示。红色细线为八面体，蓝色粗线为四面体。

所谓的四体面的中位线，即八面体的对顶线（轴线）。
显然，八面体的三条对顶线是可以任意组合的，其中没有约束关系。

17#的证明是有局限的，见我在那里的点评。

hujunhua

三角形ABC的三条中线AD, BE, CF可以构成一个三角形是有简明的几何基础的，那就是向量AD+BE+CF=0.

也就是说，那3条线不用调整方向，直接适当平移就可构成三角形。作类比推广时应该注意这个基础。

数学星空

不妨设\(A(0,0,0),B(a,0,0),C(x_1,y_1,0),P(x_0,y_0,z_0)\)则：

\((x_1-a)^2+y_1^2=b^2\)........(1)

\(x_1^2+y_1^2=c^2\).............(2)

\(x_0^2+y_0^2+z_0^2=y^2=(t-b)^2\)...........(3)

\((x_0-x_1)^2+(y_0-y_1)^2+z_0^2=x^2=(t-a)^2\)..........(4)

\((x_0-a)^2+y_0^2+z_0^2=z^2=(t-c)^2\)..............(5)

将以上消元{\(t,x_1,y_1,x,y,z\)}得到：
\(a^4-4a^3x_0-2a^2b^2+4a^2bc-2a^2c^2+4a^2x_0^2+4ab^2x_0-8abcx_0+4ac^2x_0+b^4-4b^3c+6b^2c^2-4b^2x_0^2-4b^2y_0^2-4b^2z_0^2-4bc^3+8bcx_0^2+8bcy_0^2+8bcz_0^2+c^4-4c^2x_0^2-4c^2y_0^2-4c^2z_0^2=0\)....................(6)

\(a^{12}-8a^{11}b+4a^{11}x_0+28a^{10}b^2-24a^{10}bx_0-4a^{10}c^2-2a^{10}x_0^2-6a^{10}y_0^2-8a^{10}z_0^2-56a^9b^3+56a^9b^2x_0+24a^9bc^2+24a^9bx_0^2+40a^9by_0^2+48a^9bz_0^2-8a^9c^2x_0-12a^9x_0^3-12a^9x_0y_0^2-16a^9x_0z_0^2+70a^8b^4-56a^8b^3x_0-60a^8b^2c^2-96a^8b^2x_0^2-112a^8b^2y_0^2-120a^8b^2z_0^2+24a^8bc^2x_0+56a^8bx_0^3+56a^8bx_0y_0^2+64a^8bx_0z_0^2+6a^8c^4+16a^8c^2x_0^2+8a^8c^2y_0^2+16a^8c^2z_0^2+9a^8x_0^4+34a^8x_0^2y_0^2+24a^8x_0^2z_0^2+25a^8y_0^4+40a^8y_0^2z_0^2+16a^8z_0^4-56a^7b^5+80a^7b^3c^2+184a^7b^3x_0^2+168a^7b^3y_0^2+160a^7b^3z_0^2-88a^7b^2x_0^3-88a^7b^2x_0y_0^2-80a^7b^2x_0z_0^2-24a^7bc^4-88a^7bc^2x_0^2-40a^7bc^2y_0^2-64a^7bc^2z_0^2-48a^7bx_0^4-128a^7bx_0^2y_0^2-112a^7bx_0^2z_0^2-80a^7by_0^4-144a^7by_0^2z_0^2-64a^7bz_0^4+8a^7c^2x_0^3-8a^7c^2x_0y_0^2+28a^6b^6+56a^6b^5x_0-60a^6b^4c^2-180a^6b^4x_0^2-140a^6b^4y_0^2-120a^6b^4z_0^2-80a^6b^3c^2x_0+24a^6b^3x_0^3+24a^6b^3x_0y_0^2+36a^6b^2c^4+168a^6b^2c^2x_0^2+72a^6b^2c^2y_0^2+96a^6b^2c^2z_0^2+100a^6b^2x_0^4+184a^6b^2x_0^2y_0^2+200a^6b^2x_0^2z_0^2+84a^6b^2y_0^4+184a^6b^2y_0^2z_0^2+96a^6b^2z_0^4+24a^6bc^4x_0+8a^6bc^2x_0^3+40a^6bc^2x_0y_0^2+32a^6bc^2x_0z_0^2-4a^6c^6-20a^6c^4x_0^2+4a^6c^4y_0^2-8a^6c^4z_0^2-12a^6c^2x_0^4-32a^6c^2x_0^2y_0^2-16a^6c^2x_0^2z_0^2-20a^6c^2y_0^4-16a^6c^2y_0^2z_0^2-8a^5b^7-56a^5b^6x_0+24a^5b^5c^2+72a^5b^5x_0^2+56a^5b^5y_0^2+48a^5b^5z_0^2+120a^5b^4c^2x_0+80a^5b^4x_0^3+80a^5b^4x_0y_0^2+80a^5b^4x_0z_0^2-24a^5b^3c^4-112a^5b^3c^2x_0^2-48a^5b^3c^2y_0^2-64a^5b^3c^2z_0^2-96a^5b^3x_0^4-128a^5b^3x_0^2y_0^2-160a^5b^3x_0^2z_0^2-32a^5b^3y_0^4-96a^5b^3y_0^2z_0^2-64a^5b^3z_0^4-72a^5b^2c^4x_0-96a^5b^2c^2x_0^3-96a^5b^2c^2x_0y_0^2-96a^5b^2c^2x_0z_0^2+8a^5bc^6+40a^5bc^4x_0^2-8a^5bc^4y_0^2+16a^5bc^4z_0^2+32a^5bc^2x_0^4+64a^5bc^2x_0^2y_0^2+32a^5bc^2x_0^2z_0^2+32a^5bc^2y_0^4+32a^5bc^2y_0^2z_0^2+8a^5c^6x_0+16a^5c^4x_0^3+16a^5c^4x_0y_0^2+16a^5c^4x_0z_0^2+a^4b^8+24a^4b^7x_0-4a^4b^6c^2+16a^4b^6x_0^2-8a^4b^6z_0^2-72a^4b^5c^2x_0-88a^4b^5x_0^3-88a^4b^5x_0y_0^2-64a^4b^5x_0z_0^2+6a^4b^4c^4-32a^4b^4c^2x_0^2-8a^4b^4c^2y_0^2+16a^4b^4c^2z_0^2+30a^4b^4x_0^4+44a^4b^4x_0^2y_0^2+40a^4b^4x_0^2z_0^2+14a^4b^4y_0^4+24a^4b^4y_0^2z_0^2+16a^4b^4z_0^4+72a^4b^3c^4x_0+144a^4b^3c^2x_0^3+112a^4b^3c^2x_0y_0^2+96a^4b^3c^2x_0z_0^2-4a^4b^2c^6+16a^4b^2c^4x_0^2+16a^4b^2c^4y_0^2-8a^4b^2c^4z_0^2-12a^4b^2c^2x_0^4-40a^4b^2c^2x_0^2y_0^2-28a^4b^2c^2y_0^4-32a^4b^2c^2y_0^2z_0^2-24a^4bc^6x_0-56a^4bc^4x_0^3-24a^4bc^4x_0y_0^2-32a^4bc^4x_0z_0^2+a^4c^8-8a^4c^6y_0^2-2a^4c^4x_0^4-4a^4c^4x_0^2y_0^2-8a^4c^4x_0^2z_0^2+14a^4c^4y_0^4+8a^4c^4y_0^2z_0^2-4a^3b^8x_0-24a^3b^7x_0^2-8a^3b^7y_0^2+16a^3b^6c^2x_0+24a^3b^6x_0^3+24a^3b^6x_0y_0^2+16a^3b^6x_0z_0^2+72a^3b^5c^2x_0^2+24a^3b^5c^2y_0^2+16a^3b^5x_0^4+16a^3b^5x_0^2z_0^2-16a^3b^5y_0^4-16a^3b^5y_0^2z_0^2-24a^3b^4c^4x_0-56a^3b^4c^2x_0^3-40a^3b^4c^2x_0y_0^2-32a^3b^4c^2x_0z_0^2-72a^3b^3c^4x_0^2-24a^3b^3c^4y_0^2-32a^3b^3c^2x_0^4-32a^3b^3c^2x_0^2z_0^2+32a^3b^3c^2y_0^4+32a^3b^3c^2y_0^2z_0^2+16a^3b^2c^6x_0+40a^3b^2c^4x_0^3+8a^3b^2c^4x_0y_0^2+16a^3b^2c^4x_0z_0^2+24a^3bc^6x_0^2+8a^3bc^6y_0^2+16a^3bc^4x_0^4+16a^3bc^4x_0^2z_0^2-16a^3bc^4y_0^4-16a^3bc^4y_0^2z_0^2-4a^3c^8x_0-8a^3c^6x_0^3+8a^3c^6x_0y_0^2+6a^2b^8x_0^2+2a^2b^8y_0^2+8a^2b^7x_0^3+8a^2b^7x_0y_0^2-24a^2b^6c^2x_0^2-8a^2b^6c^2y_0^2-12a^2b^6x_0^4-8a^2b^6x_0^2y_0^2-8a^2b^6x_0^2z_0^2+4a^2b^6y_0^4+8a^2b^6y_0^2z_0^2-24a^2b^5c^2x_0^3-24a^2b^5c^2x_0y_0^2+36a^2b^4c^4x_0^2+12a^2b^4c^4y_0^2+28a^2b^4c^2x_0^4+16a^2b^4c^2x_0^2y_0^2+16a^2b^4c^2x_0^2z_0^2-12a^2b^4c^2y_0^4-16a^2b^4c^2y_0^2z_0^2+24a^2b^3c^4x_0^3+24a^2b^3c^4x_0y_0^2-24a^2b^2c^6x_0^2-8a^2b^2c^6y_0^2-20a^2b^2c^4x_0^4-8a^2b^2c^4x_0^2y_0^2-8a^2b^2c^4x_0^2z_0^2+12a^2b^2c^4y_0^4+8a^2b^2c^4y_0^2z_0^2-8a^2bc^6x_0^3-8a^2bc^6x_0y_0^2+6a^2c^8x_0^2+2a^2c^8y_0^2+4a^2c^6x_0^4-4a^2c^6y_0^4-4ab^8x_0^3-4ab^8x_0y_0^2+16ab^6c^2x_0^3+16ab^6c^2x_0y_0^2-24ab^4c^4x_0^3-24ab^4c^4x_0y_0^2+16ab^2c^6x_0^3+16ab^2c^6x_0y_0^2-4ac^8x_0^3-4ac^8x_0y_0^2+b^8x_0^4+2b^8x_0^2y_0^2+b^8y_0^4-4b^6c^2x_0^4-8b^6c^2x_0^2y_0^2-4b^6c^2y_0^4+6b^4c^4x_0^4+12b^4c^4x_0^2y_0^2+6b^4c^4y_0^4-4b^2c^6x_0^4-8b^2c^6x_0^2y_0^2-4b^2c^6y_0^4+c^8x_0^4+2c^8x_0^2y_0^2+c^8y_0^4=0\)...............(7)

hujunhua

令`d=b-c`, 楼上(6)式可化简为
`\D\frac{(2x_0-a)^2}{d^2}-\frac{4(y_0^2+z_0^2)}{a^2-d^2}=1`

是一个旋转双曲面，符合几何意义。上式可由几何解释直接得到。
按几何意义，(7)式应该也能化为两个旋转双曲面方程。
因为`(x_0,y_0,z_0)`是三个旋转双曲面的交点。

数学星空

可以利用参数表示(6)及(7)

\(x_0=\frac{(b-c)(t^2+1)}{2(-t^2+1)}+\frac{1}{2a}\)

\(y_0=\frac{\sqrt{a^2-(b-c)^2}t(-s^2+1)}{(-t^2+1)(s^2+1)}\)

\(z_0=\frac{2\sqrt{a^2-(b-c)^2}ts}{(-t^2+1)(s^2+1)}\)

其中{\(s,t\)}满足下面方程：

\(4c^3at^2+8ab^3t^4+6a^2t^4b^2-4c^3t^4a-2c^2a^2+25a^4t^4+6a^2b^2-2c^2b^2-4ab^3+16t^4cba^2+4t^4b^2ca-6a^4t^2-2t^4c^2b^2-4b^3at^2-4ca^3t^2+4c^2b^2t^2+14t^4c^2a^2-2t^2b^4-8t^4bc^2a+4c^2bat^2+36ba^3t^2+t^4c^4+b^4t^4+4bc^2a+b^4+(50a^4t^4-80a^3bt^4-40a^3ct^4+12a^2b^2t^4+32a^2bct^4+28a^2c^2t^4+16ab^3t^4+8ab^2ct^4-16abc^2t^4-8ac^3t^4+2b^4t^4-4b^2c^2t^4+2c^4t^4-28a^4t^2+40a^3bt^2+24a^3ct^2-48a^2b^2t^2+16a^2bct^2+16a^2c^2t^2+24ab^3t^2+24ab^2ct^2-24abc^2t^2-24ac^3t^2+12b^4t^2-24b^2c^2t^2+12c^4t^2+2a^4-8a^3b+12a^2b^2-4a^2c^2-8ab^3+8abc^2+2b^4-4b^2c^2+2c^4)s^2+(25a^4t^4-40a^3bt^4-20a^3ct^4+6a^2b^2t^4+16a^2bct^4+14a^2c^2t^4+8ab^3t^4+4ab^2ct^4-8abc^2t^4-4ac^3t^4+b^4t^4-2b^2c^2t^4+c^4t^4-6a^4t^2+36a^3bt^2-4a^3ct^2-24a^2b^2t^2-24a^2bct^2+8a^2c^2t^2-4ab^3t^2-4ab^2ct^2+4abc^2t^2+4ac^3t^2-2b^4t^2+4b^2c^2t^2-2c^4t^2+a^4-4a^3b+6a^2b^2-2a^2c^2-4ab^3+4abc^2+b^4-2b^2c^2+c^4)s^4-20t^4ca^3-24a^2t^2b^2+a^4-24cbt^2a^2-4cat^2b^2+c^4-2t^2c^4-4a^3b+8c^2a^2t^2-40a^3t^4b=0\)

或者

\(4t^4c^3b^3+14c^4a^2t^2-16c^3a^3t^2-t^4c^2a^4-t^4c^4b^2+4t^4c^3a^3-2t^4c^5b-2t^4a^5c-6cab^4+2ab^5+6t^4a^4bc-4ca^2b^3+16ca^5t^2-2t^4b^5c-t^4b^4c^2-t^4c^4a^2-6a^4b^2t^2-6c^4ab-4t^4c^3a^2b+b^6-8ca^2b^3t^2-2c^6t^2-c^2a^4-6a^2b^2c^2+4ca^3b^2+10t^4a^2b^2c^2-2ab^5t^4-2t^4c^5a+4c^2a^3b-2c^5b+2c^2t^2b^4+25a^6-6a^6t^2-a^4b^2-30a^5c-4t^4a^2b^3c-4t^4c^3b^2a+(2a^6t^4-4a^5bt^4-4a^5ct^4-2a^4b^2t^4+12a^4bct^4-2a^4c^2t^4+8a^3b^3t^4-8a^3b^2ct^4-8a^3bc^2t^4+8a^3c^3t^4-2a^2b^4t^4-8a^2b^3ct^4+20a^2b^2c^2t^4-8a^2bc^3t^4-2a^2c^4t^4-4ab^5t^4+12ab^4ct^4-8ab^3c^2t^4-8ab^2c^3t^4+12abc^4t^4-4ac^5t^4+2b^6t^4-4b^5ct^4-2b^4c^2t^4+8b^3c^3t^4-2b^2c^4t^4-4bc^5t^4+2c^6t^4-28a^6t^2+32a^5bt^2+32a^5ct^2+36a^4b^2t^2-88a^4bct^2+36a^4c^2t^2-32a^3b^3t^2+32a^3b^2ct^2+32a^3bc^2t^2-32a^3c^3t^2-20a^2b^4t^2+48a^2b^3ct^2-56a^2b^2c^2t^2+48a^2bc^3t^2-20a^2c^4t^2+12b^6t^2-24b^5ct^2-12b^4c^2t^2+48b^3c^3t^2-12b^2c^4t^2-24bc^5t^2+12c^6t^2+50a^6-60a^5b-60a^5c-2a^4b^2+76a^4bc-2a^4c^2-8a^3b^3+8a^3b^2c+8a^3bc^2-8a^3c^3+14a^2b^4-8a^2b^3c-12a^2b^2c^2-8a^2bc^3+14a^2c^4+4ab^5-12ab^4c+8ab^3c^2+8ab^2c^3-12abc^4+4ac^5+2b^6-4b^5c-2b^4c^2+8b^3c^3-2b^2c^4-4bc^5+2c^6)s^2+(a^6t^4-2a^5bt^4-2a^5ct^4-a^4b^2t^4+6a^4bct^4-a^4c^2t^4+4a^3b^3t^4-4a^3b^2ct^4-4a^3bc^2t^4+4a^3c^3t^4-a^2b^4t^4-4a^2b^3ct^4+10a^2b^2c^2t^4-4a^2bc^3t^4-a^2c^4t^4-2ab^5t^4+6ab^4ct^4-4ab^3c^2t^4-4ab^2c^3t^4+6abc^4t^4-2ac^5t^4+b^6t^4-2b^5ct^4-b^4c^2t^4+4b^3c^3t^4-b^2c^4t^4-2bc^5t^4+c^6t^4-6a^6t^2+16a^5bt^2+16a^5ct^2-6a^4b^2t^2-28a^4bct^2-6a^4c^2t^2-16a^3b^3t^2+16a^3b^2ct^2+16a^3bc^2t^2-16a^3c^3t^2+14a^2b^4t^2-8a^2b^3ct^2-12a^2b^2c^2t^2-8a^2bc^3t^2+14a^2c^4t^2-2b^6t^2+4b^5ct^2+2b^4c^2t^2-8b^3c^3t^2+2b^2c^4t^2+4bc^5t^2-2c^6t^2+25a^6-30a^5b-30a^5c-a^4b^2+38a^4bc-a^4c^2-4a^3b^3+4a^3b^2c+4a^3bc^2-4a^3c^3+7a^2b^4-4a^2b^3c-6a^2b^2c^2-4a^2bc^3+7a^2c^4+2ab^5-6ab^4c+4ab^3c^2+4ab^2c^3-6abc^4+2ac^5+b^6-2b^5c-b^4c^2+4b^3c^3-b^2c^4-2bc^5+c^6)s^4+7a^2b^4-2b^5c-b^4c^2-30a^5b+7c^4a^2-4c^3a^3-4c^3ba^2+6t^4bc^4a+14a^2t^2b^4-8c^3bt^2a^2-28ca^4t^2b-4t^4a^3b^2c+6t^4b^4ca-4t^4c^2a^3b+t^4c^6+2c^5a+38ca^4b+4c^5t^2b-16t^2a^3b^3+16a^5bt^2+4c^3b^2a+4c^2b^3a-4t^4c^2b^3a+4cb^5t^2+a^6t^4-12c^2a^2t^2b^2-4a^3b^3+4c^3b^3-c^4b^2+16c^2ba^3t^2-8c^3t^2b^3+b^6t^4-2b^6t^2+2b^2t^2c^4-a^2t^4b^4-a^4t^4b^2-2a^5t^4b+4a^3t^4b^3-6c^2a^4t^2+16ca^3b^2t^2+c^6=0\)

数学星空

对于\(a=3,b=4,c=5\)我们可以将(6),(7)简化为下面方程：

\(8x_0^2-y_0^2-z_0^2-24x_0+16=0\)

\((8x_0^2+24x_0y_0+15y_0^2-z_0^2-72x_0-96y_0+144)(8x_0^2-24x_0y_0+15y_0^2-z_0^2-72x_0+96y_0+144)=0\)

画图得到：

陈九章

谢谢星空老师和hujunhua老师！