三角形内三个四边形的内切圆

mathe

作出一个比较一般情况的三等圆情况，应该不满足星空的条件

wayne

对于给定的三角形$ABC$，可以唯一的做出三条内角平分线，交于一点。 然后给定半径$r$,分别在三条角平分线上取内切的小圆圆心，做切两边的小圆， 然后，继续取三个小切圆两两之间的内公切线，计算六条内公切线之间 共点状况，这样做 会不会更加简洁一点。

mathe

mathe 发表于 2018-12-27 08:06
作出一个比较一般情况的三等圆情况，应该不满足星空的条件

P点是不唯一的
可以使用wayne的方法作图，用几何画板，对于随便选一个三角形，选择一个圆的半径（可以动态改变），就可以作出三个圆和它们两两间的内公切线。
然后改变半径长度，如下图可以对相同的三角形得到至少三个不同的解（黑色的点为符合条件的解），其中有两个是三个圆有公切线的特殊情况

wayne

P点是不唯一的,而且根据作图的内公切线的分组的特点，好像可以得到结论：
1） 符合条件的P点是不同颜色的三条内公切线相交，也就是说其个数是2个，或者1个(重合)，

2） 这两个符合条件的点的连线的中点 刚好就是 三个圆心组成的三角形的重心[中线交点]。




我是用Geogebra作图的，作图文件也分享出来：

数学星空

关于三角形ABC内存在点P分成三个等内切圆需满足下列6个方程，涉及6个未知数\({x,y,z,x0,y0,r0}\),其中\(PD=x,PE=y,PF=z,P(x0,y0)\),内切圆半径为r0,三角形ABC面积为s

\(a^5-3a^4b-3a^4c-6a^4x+4a^4y+4a^4z+2a^3b^2+8a^3bc+12a^3bx-8a^3by-8a^3bz+2a^3c^2+12a^3cx-8a^3cy-8a^3cz+12a^3x^2-16a^3xy-16a^3xz+4a^3y^2+12a^3yz+4a^3z^2+2a^2b^3-6a^2b^2c-6a^2bc^2-24a^2bcx+16a^2bcy+16a^2bcz-12a^2bx^2+16a^2bxy+16a^2bxz-4a^2by^2-12a^2byz-4a^2bz^2+2a^2c^3-12a^2cx^2+16a^2cxy+16a^2cxz-4a^2cy^2-12a^2cyz-4a^2cz^2-8a^2x^3+16a^2x^2y+16a^2x^2z-8a^2xy^2-24a^2xyz-8a^2xz^2+8a^2y^2z+8a^2yz^2-3ab^4-12ab^3x+8ab^3y+8ab^3z+6ab^2c^2+12ab^2cx-8ab^2cy-8ab^2cz-12ab^2x^2+16ab^2xy+16ab^2xz-4ab^2y^2-12ab^2yz-4ab^2z^2+12abc^2x-8abc^2y-8abc^2z+16abcr0^2+24abcx^2-32abcxy-32abcxz+8abcy^2+24abcyz+8abcz^2-3ac^4-12ac^3x+8ac^3y+8ac^3z-12ac^2x^2+16ac^2xy+16ac^2xz-4ac^2y^2-12ac^2yz-4ac^2z^2+b^5+b^4c+6b^4x-4b^4y-4b^4z-2b^3c^2+12b^3x^2-16b^3xy-16b^3xz+4b^3y^2+12b^3yz+4b^3z^2-2b^2c^3-12b^2c^2x+8b^2c^2y+8b^2c^2z-16b^2cr0^2-12b^2cx^2+16b^2cxy+16b^2cxz-4b^2cy^2-12b^2cyz-4b^2cz^2+8b^2x^3-16b^2x^2y-16b^2x^2z+8b^2xy^2+24b^2xyz+8b^2xz^2-8b^2y^2z-8b^2yz^2+bc^4-16bc^2r0^2-12bc^2x^2+16bc^2xy+16bc^2xz-4bc^2y^2-12bc^2yz-4bc^2z^2-32bcr0^2x-16bcx^3+32bcx^2y+32bcx^2z-16bcxy^2-48bcxyz-16bcxz^2+16bcy^2z+16bcyz^2+c^5+6c^4x-4c^4y-4c^4z+12c^3x^2-16c^3xy-16c^3xz+4c^3y^2+12c^3yz+4c^3z^2+8c^2x^3-16c^2x^2y-16c^2x^2z+8c^2xy^2+24c^2xyz+8c^2xz^2-8c^2y^2z-8c^2yz^2+16a^2r0s-32abr0s-32acr0s-64ar0sx+32ar0sy+32ar0sz+16b^2r0s+32bcr0s+64br0sx-32br0sy-32br0sz+16c^2r0s+64cr0sx-32cr0sy-32cr0sz+64r0sx^2-64r0sxy-64r0sxz+64r0syz=0\)

\(a^5-3a^4b+a^4c-4a^4x+6a^4y-4a^4z+2a^3b^2+8a^3bx-12a^3by+8a^3bz-2a^3c^2+4a^3x^2-16a^3xy+12a^3xz+12a^3y^2-16a^3yz+4a^3z^2+2a^2b^3-6a^2b^2c+6a^2bc^2-8a^2bcx+12a^2bcy-8a^2bcz-4a^2bx^2+16a^2bxy-12a^2bxz-12a^2by^2+16a^2byz-4a^2bz^2-2a^2c^3+8a^2c^2x-12a^2c^2y+8a^2c^2z-16a^2cr0^2-4a^2cx^2+16a^2cxy-12a^2cxz-12a^2cy^2+16a^2cyz-4a^2cz^2+8a^2x^2y-8a^2x^2z-16a^2xy^2+24a^2xyz-8a^2xz^2+8a^2y^3-16a^2y^2z+8a^2yz^2-3ab^4+8ab^3c-8ab^3x+12ab^3y-8ab^3z-6ab^2c^2+16ab^2cx-24ab^2cy+16ab^2cz-4ab^2x^2+16ab^2xy-12ab^2xz-12ab^2y^2+16ab^2yz-4ab^2z^2-8abc^2x+12abc^2y-8abc^2z+16abcr0^2+8abcx^2-32abcxy+24abcxz+24abcy^2-32abcyz+8abcz^2+ac^4-16ac^2r0^2-4ac^2x^2+16ac^2xy-12ac^2xz-12ac^2y^2+16ac^2yz-4ac^2z^2-32acr0^2y-16acx^2y+16acx^2z+32acxy^2-48acxyz+16acxz^2-16acy^3+32acy^2z-16acyz^2+b^5-3b^4c+4b^4x-6b^4y+4b^4z+2b^3c^2-8b^3cx+12b^3cy-8b^3cz+4b^3x^2-16b^3xy+12b^3xz+12b^3y^2-16b^3yz+4b^3z^2+2b^2c^3-4b^2cx^2+16b^2cxy-12b^2cxz-12b^2cy^2+16b^2cyz-4b^2cz^2-8b^2x^2y+8b^2x^2z+16b^2xy^2-24b^2xyz+8b^2xz^2-8b^2y^3+16b^2y^2z-8b^2yz^2-3bc^4+8bc^3x-12bc^3y+8bc^3z-4bc^2x^2+16bc^2xy-12bc^2xz-12bc^2y^2+16bc^2yz-4bc^2z^2+c^5-4c^4x+6c^4y-4c^4z+4c^3x^2-16c^3xy+12c^3xz+12c^3y^2-16c^3yz+4c^3z^2+8c^2x^2y-8c^2x^2z-16c^2xy^2+24c^2xyz-8c^2xz^2+8c^2y^3-16c^2y^2z+8c^2yz^2+16a^2r0s-32abr0s+32acr0s-32ar0sx+64ar0sy-32ar0sz+16b^2r0s-32bcr0s+32br0sx-64br0sy+32br0sz+16c^2r0s-32cr0sx+64cr0sy-32cr0sz-64r0sxy+64r0sxz+64r0sy^2-64r0syz=0\)

\(a^5+a^4b-3a^4c-4a^4x-4a^4y+6a^4z-2a^3b^2+2a^3c^2+8a^3cx+8a^3cy-12a^3cz+4a^3x^2+12a^3xy-16a^3xz+4a^3y^2-16a^3yz+12a^3z^2-2a^2b^3+6a^2b^2c+8a^2b^2x+8a^2b^2y-12a^2b^2z-6a^2bc^2-8a^2bcx-8a^2bcy+12a^2bcz-16a^2br0^2-4a^2bx^2-12a^2bxy+16a^2bxz-4a^2by^2+16a^2byz-12a^2bz^2+2a^2c^3-4a^2cx^2-12a^2cxy+16a^2cxz-4a^2cy^2+16a^2cyz-12a^2cz^2-8a^2x^2y+8a^2x^2z-8a^2xy^2+24a^2xyz-16a^2xz^2+8a^2y^2z-16a^2yz^2+8a^2z^3+ab^4-6ab^2c^2-8ab^2cx-8ab^2cy+12ab^2cz-16ab^2r0^2-4ab^2x^2-12ab^2xy+16ab^2xz-4ab^2y^2+16ab^2yz-12ab^2z^2+8abc^3+16abc^2x+16abc^2y-24abc^2z+16abcr0^2+8abcx^2+24abcxy-32abcxz+8abcy^2-32abcyz+24abcz^2-32abr0^2z+16abx^2y-16abx^2z+16abxy^2-48abxyz+32abxz^2-16aby^2z+32abyz^2-16abz^3-3ac^4-8ac^3x-8ac^3y+12ac^3z-4ac^2x^2-12ac^2xy+16ac^2xz-4ac^2y^2+16ac^2yz-12ac^2z^2+b^5-3b^4c-4b^4x-4b^4y+6b^4z+2b^3c^2+8b^3cx+8b^3cy-12b^3cz+4b^3x^2+12b^3xy-16b^3xz+4b^3y^2-16b^3yz+12b^3z^2+2b^2c^3-4b^2cx^2-12b^2cxy+16b^2cxz-4b^2cy^2+16b^2cyz-12b^2cz^2-8b^2x^2y+8b^2x^2z-8b^2xy^2+24b^2xyz-16b^2xz^2+8b^2y^2z-16b^2yz^2+8b^2z^3-3bc^4-8bc^3x-8bc^3y+12bc^3z-4bc^2x^2-12bc^2xy+16bc^2xz-4bc^2y^2+16bc^2yz-12bc^2z^2+c^5+4c^4x+4c^4y-6c^4z+4c^3x^2+12c^3xy-16c^3xz+4c^3y^2-16c^3yz+12c^3z^2+8c^2x^2y-8c^2x^2z+8c^2xy^2-24c^2xyz+16c^2xz^2-8c^2y^2z+16c^2yz^2-8c^2z^3+16a^2r0s+32abr0s-32acr0s-32ar0sx-32ar0sy+64ar0sz+16b^2r0s-32bcr0s-32br0sx-32br0sy+64br0sz+16c^2r0s+32cr0sx+32cr0sy-64cr0sz+64r0sxy-64r0sxz-64r0syz+64r0sz^2=0\)

\(a^2-2ab+2ac-4ax0+4ay-4az+b^2-2bc+4bx0-4by+4bz+c^2-4cx0+4cy-4cz-4x^2+4x0^2-8x0y+8x0z+4y^2-8yz+4y0^2+4z^2=0\)

\(2a^4-3a^3b-2a^3c-4a^3x-2a^3x0+4a^3z+2a^2bc+4a^2bx+6a^2bx0-4a^2bz-2a^2c^2+2a^2cx0+4a^2xx0-4a^2x0z+ab^3-2ab^2x0-3abc^2-4abcx+4abcz-4abx^2+8abxz-4abx0^2+4aby^2-4aby0^2-4abz^2+2ac^3+4ac^2x+2ac^2x0-4ac^2z-2b^3x0+2b^2cx0+4b^2xx0-4b^2x0z+2bc^2x0-2c^3x0-4c^2xx0+4c^2x0z+8asy0+8bsy0-8csy0-16sxy0+16sy0z=0\)

\(a^3c-2a^3x0-2a^2bc+2a^2bx0+2a^2c^2-4a^2cx-2a^2cx0+4a^2cy+4a^2xx0-4a^2x0y+ab^2c+2ab^2x0-2abc^2+4abcx-4abcy+ac^3-4ac^2x-2ac^2x0+4ac^2y+4acx^2-8acxy+4acx0^2+4acy^2+4acy0^2-4acz^2-2b^3x0+2b^2cx0-4b^2xx0+4b^2x0y+2bc^2x0-2c^3x0+4c^2xx0-4c^2x0y-8asy0+8bsy0-8csy0+16sxy0-16syy0=0\)

下面方程可以检验上面的解是否位于三角形内

\(ar0+br0+cr0+2r0x+2r0y+2r0z-2s=0\)

数学星空

下面是根据上面的方程计算的实例：

1. \(a = 7, b = 8, c = 9\) 可以得到

\({r0 = 1.557189142, s = 26.83281572, x = 2.609051087, x0 = 5.498057642, y = 2.609051100, y0 = 1.702161582,z = 3.588313240}\)




\({r0 = 1.348573921, s = 26.83281572, x = 3.103534622, x0 = 4.212869410, y = 2.130953391, y0 = 3.012887094,  z = 2.662690260}\)




2.\(a = 8, b = 8, c = 9\)时未找到合适的解？？


3.\(a = 8, b = 9, c = 9\)可以得到

\({r0 = 1.531128874, s = 32.24903100, x = 2.033741334, x0 = 4.000000000,  y = 3.014258212, y0 = 2.033741334,z = 3.014258212}\)



\({r0 = 1.531128874, s = 32.24903100, x = 3.051670350, x0 = 4.000000000, y = 2.477478319, y0 = 3.051670350, z = 2.477478319}\)



\({r0 = 1.531128874, s = 32.24903100, x = 3.085908626, x0 = 4.000000000, y = 2.488174239, y0 = 3.085908626, z = 2.488174239}\)

hujunhua

数学星空 发表于 2018-12-26 21:10
对于存在三个等内切圆时,且EF平行于底边BC时三角形三边需满足关系

\(5a^4-2a^2b^2-2a^2c^2-3b^4+6b^2c^2 ...

化成角度关系式更简明：\[\tan B\tan C=4\tag1\] 这表明固定顶点 `B:=(0,-1)` 和 `C:=(0,1)`，那么顶点 `A` 就在椭圆\[x^2+y^2/4=1\tag2\]上。这是因为`(1)`式表明 `BC` 边上的高 `AH` 的中点 `M`在以 `BC` 为直径的圆上。

下面给出 `P` 位置作图。

`\triangle A_1B_1C_1`与`\triangle ABC`位似，位似中心即内心 `I`. 在此由位似关系中，`P` 与 `M`相对应，故 `P` 在连线 `IM`上。
`\triangle AFE` 与 `\triangle ABC` 位似，位似中心为顶点 `A`. 在此由位似关系中，`P` 与 `T`相对应，故 `P` 在连线 `AT`上。
所以 `P`是 `AT` 与 `IM` 的交点。

数学星空

4.\(a = 8, b = 9, c = 10\)时得到

  \({ r0 = 1.535826535, s = 34.19703935, x = 3.430060840, x0 = 4.713346513,  y = 2.395556848, y0 = 3.345171331, z = 2.940594227}\)


  
\({ r0 = 1.535826538, s = 34.19703935, x = 2.020380262, x0 = 4.204632604, y = 3.448584501, y0 = 1.970378501,  z = 3.297247334}\)





5.\(a = 8, b = 9, c = 11\)时得到

\({r0 = 1.505143129, s = 35.49647870, x = 2.026681074, x0 = 4.442147323,y = 3.882063456, y0 = 1.876672520,  z = 3.674712613}\)



\({r0 = 1.505143137, s = 35.49647870, x = 3.820472274, x0 = 5.422398236, y = 2.371453446, y0 = 3.537692858,  z = 3.391531426}\)



6.\(a = 8, b = 9, c = 12\)时得到

\({r0 = 1.444105426, s = 35.99913192, x = 2.040639136, x0 = 4.686943432,y = 4.302762090, y0 = 1.761098380, z = 4.084925561}\)

wayne

不用解析几何，可以直接列关系式的。这样少了坐标点引入的大量未知数，直接揭示几何元素的量的关系。
设三角形$ABC$三边边长分别是$a,b,c$, 三个内切圆的半径是$r$,圆心分别是$O_1,O_2,O_3$，分别靠近三个角$A,B,C$，
$O_2,O_3$的中点是$D$,  $O_1,O_3$的中点是$E$,  $O_1,O_2$的中点是$F$,那么设满足条件的点是$P$, 容易得到关系：\[
O_1O_2 = c-r\cot A-r\cot B
\\O_1O_3 = b-r\cot A-r\cot C
\\O_2O_3 = a-r\cot B-r\cot C
\]根据内公切线的关系，得到：\[
(PO_2^2-PO_3^2)^2 = PD^2 \cdot(O_2O_3^2-4r^2)
\\ (PO_1^2-PO_3^2)^2 = PE^2 \cdot(O_1O_3^2-4r^2)
\\ (PO_1^2-PO_2^2)^2 = PF^2 \cdot(O_1O_2^2-4r^2)\]

数学星空

为了方便计算，我们可以得到更简洁的形式，大曲边三角形的代数曲线：

\(-2a(a+b-c)(a-b+c)(-b-c+a)(2a^3-2a^2b+2a^2c-ab^2-2abc+3ac^2+b^3-b^2c-bc^2+c^3)xy^2+(2(a+b-c))(a-b+c)(-b-c+a)(2a^3+ab^2-2abc+ac^2+b^3-b^2c-bc^2+c^3)x^2y^2+a^2(c+a+b)(-b-c+a)(a+b-c)(a-b+c)^3y^2+(2a^3-ab^2+2abc-ac^2-b^3+b^2c+bc^2-c^3)^2y^4+a^2(a+b-c)^2(a-b+c)^2(-b-c+a)^2x^2-2a(a+b-c)^2(a-b+c)^2(-b-c+a)^2x^3+(a+b-c)^2(a-b+c)^2(-b-c+a)^2x^4=0\)


\(-4a(a+b-c)(a-b+c)(-b-c+a)(a^5+3a^4b-3a^4c-2a^3b^2-8a^3bc+2a^3c^2+2a^2b^3-10a^2b^2c+6a^2bc^2+2a^2c^3-3ab^4+6ab^2c^2-3ac^4-b^5+b^4c+2b^3c^2-2b^2c^3-bc^4+c^5)(a^2-2ab-2ac-b^2+c^2)xy^2+14a^{12}bc+(a^5+3a^4b-3a^4c+6a^3b^2-8a^3bc+2a^3c^2-6a^2b^3-2a^2b^2c+6a^2bc^2+2a^2c^3-3ab^4+6ab^2c^2-3ac^4-b^5+b^4c+2b^3c^2-2b^2c^3-bc^4+c^5)^2y^4-6a^{13}c-40a^{10}b^3c-2a^{13}b+(2(a+b-c))(a-b+c)(-b-c+a)(a^7-a^6b-5a^6c-7a^5b^2-2a^5bc+9a^5c^2+31a^4b^3+21a^4b^2c+17a^4bc^2-5a^4c^3+15a^3b^4+28a^3b^3c-10a^3b^2c^2-28a^3bc^3-5a^3c^4+17a^2b^5+9a^2b^4c-34a^2b^3c^2-18a^2b^2c^3+17a^2bc^4+9a^2c^5+7ab^6-2ab^5c-19ab^4c^2+4ab^3c^3+17ab^2c^4-2abc^5-5ac^6+b^7-b^6c-3b^5c^2+3b^4c^3+3b^3c^4-3b^2c^5-bc^6+c^7)x^2y^2+(a-b+c)^2(-b-c+a)^2(a+b-c)^2(a^2-4ab-2ac-b^2+c^2)^2x^4+12a^5b^4c^5+36a^5b^5c^4-2a^4b^9c-40a^6bc^7+12a^6b^2c^6+72a^6b^3c^5-66a^6b^4c^4-24a^5b^6c^3-8a^5b^7c^2-4a(a^2-2ab-2ac-b^2+c^2)(a^2-4ab-2ac-b^2+c^2)(a+b-c)^2(a-b+c)^2(-b-c+a)^2x^3+110a^8b^2c^4-24a^6b^5c^3+13a^{12}c^2-3a^{12}b^2+10a^5b^8c+56a^7bc^6+104a^7b^4c^3-24a^7b^5c^2-24a^7b^6c-72a^9b^2c^3+44a^6b^6c^2-8a^6b^7c-28a^8bc^5+72a^9b^3c^2-72a^7b^2c^5-40a^7b^3c^4+36a^8b^5c+56a^{10}bc^3+12a^{10}b^2c^2+2a^2(a-b+c)(a^5+3a^4b-3a^4c-2a^3b^2-8a^3bc+2a^3c^2+2a^2b^3-10a^2b^2c+6a^2bc^2+2a^2c^3-3ab^4+6ab^2c^2-3ac^4-b^5+b^4c+2b^3c^2-2b^2c^3-bc^4+c^5)(-b-c+a)^2(a+b-c)^2y^2-4a^3(a^2-2ab-2ac-b^2+c^2)(a-b+c)^2(-b-c+a)^3(a+b-c)^3x+2a^2(3a^4-12a^3b-12a^3c+2a^2b^2+24a^2bc+18a^2c^2+12ab^3+12ab^2c-12abc^2-12ac^3+3b^4-6b^2c^2+3c^4)(a+b-c)^2(a-b+c)^2(-b-c+a)^2x^2-40a^8b^3c^3-66a^8b^4c^2+a^{14}+8a^5b^2c^7-28a^9bc^4+12a^9b^4c-6a^5c^9-2a^5b^9-8a^7c^7-40a^{11}bc^2+8a^{11}b^2c-2a^4bc^9-3a^4b^2c^8+8a^7b^7+28a^9c^5-12a^9b^5-8a^{11}c^3+8a^{11}b^3+a^4c^{10}+a^4b^{10}+13a^6c^8-3a^6b^8+14a^5bc^8-40a^5b^3c^6-14a^8c^6+2a^8b^6+8a^4b^3c^7+2a^4b^4c^6-12a^4b^5c^5-14a^{10}c^4+2a^{10}b^4+2a^4b^6c^4+8a^4b^7c^3-3a^4b^8c^2=0\)



\(8a^2c(a+b-c)(a-b+c)(-b-c+a)(a^5-3a^4b-a^4c+2a^3b^2-4a^3bc-14a^3c^2+2a^2b^3+10a^2b^2c-2a^2bc^2-10a^2c^3-3ab^4-4ab^3c+10ab^2c^2+4abc^3-7ac^4+b^5-b^4c-2b^3c^2+2b^2c^3+bc^4-c^5)xy^2+(2(a+b-c))(a-b+c)(-b-c+a)(a^7-5a^6b-a^6c+9a^5b^2-2a^5bc-7a^5c^2-5a^4b^3+17a^4b^2c+21a^4bc^2+31a^4c^3-5a^3b^4-28a^3b^3c-10a^3b^2c^2+28a^3bc^3+15a^3c^4+9a^2b^5+17a^2b^4c-18a^2b^3c^2-34a^2b^2c^3+9a^2bc^4+17a^2c^5-5ab^6-2ab^5c+17ab^4c^2+4ab^3c^3-19ab^2c^4-2abc^5+7ac^6+b^7-b^6c-3b^5c^2+3b^4c^3+3b^3c^4-3b^2c^5-bc^6+c^7)x^2y^2+16a^4c^2(c+a+b)(-b-c+a)(a+b-c)(a-b+c)^3y^2+(a^5-3a^4b+3a^4c+2a^3b^2-8a^3bc+6a^3c^2+2a^2b^3+6a^2b^2c-2a^2bc^2-6a^2c^3-3ab^4+6ab^2c^2-3ac^4+b^5-b^4c-2b^3c^2+2b^2c^3+bc^4-c^5)^2y^4+16a^4c^2(a+b-c)^2(a-b+c)^2(-b-c+a)^2x^2+8a^2c(a^2-2ab-4ac+b^2-c^2)(a+b-c)^2(a-b+c)^2(-b-c+a)^2x^3+(a+b-c)^2(a-b+c)^2(-b-c+a)^2(a^2-2ab-4ac+b^2-c^2)^2x^4=0\)