三角形正负等角中心间距

数学星空

可以消元得到轨迹方程：

9*a^16-108*a^14*b^2-36*a^14*x^2-60*a^14*y^2+486*a^12*b^4+348*a^12*b^2*x^2-60*a^12*b^2*y^2+54*a^12*x^4+156*a^12*x^2*y^2+118*a^12*y^4-1044*a^10*b^6-1308*a^10*b^4*x^2+732*a^10*b^4*y^2-396*a^10*b^2*x^4-376*a^10*b^2*x^2*y^2-268*a^10*b^2*y^4-36*a^10*x^6-132*a^10*x^4*y^2-156*a^10*x^2*y^4-60*a^10*y^6+1161*a^8*b^8+2580*a^8*b^6*x^2-1236*a^8*b^6*y^2+1174*a^8*b^4*x^4-356*a^8*b^4*x^2*y^2+150*a^8*b^4*y^4+180*a^8*b^2*x^6+532*a^8*b^2*x^4*y^2+524*a^8*b^2*x^2*y^4+172*a^8*b^2*y^6+9*a^8*x^8+36*a^8*x^6*y^2+54*a^8*x^4*y^4+36*a^8*x^2*y^6+9*a^8*y^8-648*a^6*b^10-2832*a^6*b^8*x^2+816*a^6*b^8*y^2-1888*a^6*b^6*x^4+1728*a^6*b^6*x^2*y^2+32*a^6*b^6*y^4-368*a^6*b^4*x^6-912*a^6*b^4*x^4*y^2-720*a^6*b^4*x^2*y^4-176*a^6*b^4*y^6-24*a^6*b^2*x^8-96*a^6*b^2*x^6*y^2-144*a^6*b^2*x^4*y^4-96*a^6*b^2*x^2*y^6-24*a^6*b^2*y^8+144*a^4*b^12+1632*a^4*b^10*x^2-192*a^4*b^10*y^2+1824*a^4*b^8*x^4-1664*a^4*b^8*x^2*y^2-32*a^4*b^8*y^4+352*a^4*b^6*x^6+768*a^4*b^6*x^4*y^2+480*a^4*b^6*x^2*y^4+64*a^4*b^6*y^6+16*a^4*b^4*x^8+64*a^4*b^4*x^6*y^2+96*a^4*b^4*x^4*y^4+64*a^4*b^4*x^2*y^6+16*a^4*b^4*y^8-384*a^2*b^12*x^2-1024*a^2*b^10*x^4+512*a^2*b^10*x^2*y^2-128*a^2*b^8*x^6-256*a^2*b^8*x^4*y^2-128*a^2*b^8*x^2*y^4+256*b^12*x^4=0


\(9a^{16}-108a^{14}b^2-36a^{14}x^2-60a^{14}y^2+486a^{12}b^4+348a^{12}b^2x^2-60a^{12}b^2y^2+54a^{12}x^4+156a^{12}x^2y^2+118a^{12}y^4-1044a^{10}b^6-1308a^{10}b^4x^2+732a^{10}b^4y^2-396a^{10}b^2x^4-376a^{10}b^2x^2y^2-268a^{10}b^2y^4-36a^{10}x^6-132a^{10}x^4y^2-156a^{10}x^2y^4-60a^{10}y^6+1161a^8b^8+2580a^8b^6x^2-1236a^8b^6y^2+1174a^8b^4x^4-356a^8b^4x^2y^2+150a^8b^4y^4+180a^8b^2x^6+532a^8b^2x^4y^2+524a^8b^2x^2y^4+172a^8b^2y^6+9a^8x^8+36a^8x^6y^2+54a^8x^4y^4+36a^8x^2y^6+9a^8y^8-648a^6b^{10}-2832a^6b^8x^2+816a^6b^8y^2-1888a^6b^6x^4+1728a^6b^6x^2y^2+32a^6b^6y^4-368a^6b^4x^6-912a^6b^4x^4y^2-720a^6b^4x^2y^4-176a^6b^4y^6-24a^6b^2x^8-96a^6b^2x^6y^2-144a^6b^2x^4y^4-96a^6b^2x^2y^6-24a^6b^2y^8+144a^4b^{12}+1632a^4b^{10}x^2-192a^4b^{10}y^2+1824a^4b^8x^4-1664a^4b^8x^2y^2-32a^4b^8y^4+352a^4b^6x^6+768a^4b^6x^4y^2+480a^4b^6x^2y^4+64a^4b^6y^6+16a^4b^4x^8+64a^4b^4x^6y^2+96a^4b^4x^4y^4+64a^4b^4x^2y^6+16a^4b^4y^8-384a^2b^{12}x^2-1024a^2b^{10}x^4+512a^2b^{10}x^2y^2-128a^2b^8x^6-256a^2b^8x^4y^2-128a^2b^8x^2y^4+256b^{12}x^4=0\)

例如：取 \(a=5,b=3\) 得到

\(950625x^8+3802500x^6y^2+5703750x^4y^4+3802500x^2y^6+950625y^8-45115200x^6-265070400x^4y^2-394795200x^2y^4-174840000y^6+123434496x^4+7390387200x^2y^2+10233280000y^4+9772646400x^2-194426880000y^2+44605440000=0\)

我们可以取点
\(a = 5, b = 3, c = 4, x = \frac{836800}{196729}-\frac{256000}{196729}\sqrt{3}, x_0 = 3, y =\frac{354560}{196729}-\frac{372052}{590187}\sqrt{3}, y_0 =\frac{12}{5}\)

画图得到

陈九章

今日学校放月假，我行吟在二教大楼前坪，踏响《荷塘月色》的优美旋律：
剪一段时光缓缓流淌，流进了月色中微微荡漾，弹一首小荷淡淡的香，美丽的琴音就落在我身旁，
萤火虫点亮夜的星光，谁为我添一件梦的衣裳，推开那扇心窗远远地望，谁采下那一朵昨日的忧伤，
我像只鱼儿在你的荷塘，只为和你守候那皎洁月光......
忽然看到平时忙于事业、难得上网的数学星空老师的论帖，非常高兴，撰联一副欢迎：
仰望星空，数学招来云外客；
行吟月夜，难题引出洞中仙。

补充内容 (2020-6-12 20:40):
据《湖南日报》特约通讯员校办周主任说：这里原是一口长满荷花的大池塘------鲤鱼塘，叶动无穷碧，花开别样红。

陈九章

陈九章

很有趣的问题，找到了：

陈九章

想起来了，这个问题，我以前考虑过，如下第3个结论，推导过程，现在没有印象了，请老师审阅、赐教！

陈九章

数学星空

椭圆内接N边形周长最大值
https://bbs.emath.ac.cn/thread-3740-1-1.html

陈九章

这个结论，请星空老师审阅，并赐教！

补充内容 (2023-4-28 16:50):
这个结论有误。

葡萄糖

陈九章 发表于 2020-6-12 20:35
很有趣的问题，找到了：

\(\,\dfrac{x^{2}}{\left(\left(\frac{a^{2}-b^{2}}{a^{2}+3b^{2}}\right)a\right)^{2}}+\dfrac{y^{2}}{\left(\left(\frac{a^{2}-b^{2}}{3a^{2}+b^{2}}\right)b\right)^{2}}=1\,\)
好奇，轨迹上点坐标与正三角形边长的关系

数学星空

对于内接椭圆\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)的正三角形问题：

其边长设为\(L\),便于消元计算设正三角的三个顶点坐标为

\([\frac{a(1-m^2)}{1+m^2},\frac{2bm}{1+m^2}],[\frac{a(1-n^2)}{1+n^2},\frac{2bn}{1+n^2}],[\frac{a(1-p^2)}{1+p^2},\frac{2bp}{1+p^2}]\)

正三角形的中心\([x_0,y_0]\),其中正三角形的一个顶点\(x=\frac{a(1-m^2)}{1+m^2},y=\frac{2bm}{1+m^2}\)

\(-L^2m^4n^4-2L^2m^4n^2-2L^2m^2n^4+4b^2m^4n^2-8b^2m^3n^3+4b^2m^2n^4-L^2m^4-4L^2m^2n^2-L^2n^4+4a^2m^4-8a^2m^2n^2+4a^2n^4-8b^2m^3n+16b^2m^2n^2-8b^2mn^3-2L^2m^2-2L^2n^2+4b^2m^2-8b^2mn+4b^2n^2-L^2=0\)

\(-L^2n^4p^4-2L^2n^4p^2-2L^2n^2p^4+4b^2n^4p^2-8b^2n^3p^3+4b^2n^2p^4-L^2n^4-4L^2n^2p^2-L^2p^4+4a^2n^4-8a^2n^2p^2+4a^2p^4-8b^2n^3p+16b^2n^2p^2-8b^2np^3-2L^2n^2-2L^2p^2+4b^2n^2-8b^2np+4b^2p^2-L^2=0\)

\(-L^2m^4p^4-2L^2m^4p^2-2L^2m^2p^4+4b^2m^4p^2-8b^2m^3p^3+4b^2m^2p^4-L^2m^4-4L^2m^2p^2-L^2p^4+4a^2m^4-8a^2m^2p^2+4a^2p^4-8b^2m^3p+16b^2m^2p^2-8b^2mp^3-2L^2m^2-2L^2p^2+4b^2m^2-8b^2mp+4b^2p^2-L^2=0\)

\(3am^2n^2p^2+3m^2n^2p^2x_0+am^2n^2+am^2p^2+an^2p^2+3m^2n^2x_0+3m^2p^2x_0+3n^2p^2x_0-am^2-an^2-ap^2+3m^2x_0+3n^2x_0+3p^2x_0-3a+3x_0=0\)

\(3m^2n^2p^2y_0-2bm^2n^2p-2bm^2np^2-2bmn^2p^2+3m^2n^2y_0+3m^2p^2y_0+3n^2p^2y_0-2bm^2n-2bm^2p-2bmn^2-2bmp^2-2bn^2p-2bnp^2+3m^2y_0+3n^2y_0+3p^2y_0-2bm-2bn-2bp+3y_0=0\)

正三角形中心轨迹方程：

\(-a^2(3a^2+b^2)^2y_0^2-b^2(a^2+3b^2)^2x_0^2+a^2b^2(a-b)^2(a+b)^2=0\)

椭圆\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)内接正三角的顶点\([x,y]\)与边长L的关系\(x=a\cos(\theta),y=b\sin(\theta)\)

\((81a^{12}+378a^{10}b^2+63a^8b^4-1044a^6b^6+63a^4b^8+378a^2b^{10}+81b^{12})L^6+(648\cos(2\theta)a^{12}b^2+216\cos(2\theta)a^{10}b^4+2256\cos(2\theta)a^8b^6-2256\cos(2\theta)a^6b^8-216\cos(2\theta)a^4b^{10}-648\cos(2\theta)a^2b^{12}-648a^{12}b^2-3816a^{10}b^4+4464b^6a^8+4464b^8a^6-3816b^{10}a^4-648a^2b^{12})L^4+(864\cos(4\theta)a^{12}b^4-1920\cos(4\theta)a^{10}b^6+2112\cos(4\theta)a^8b^8-1920\cos(4\theta)a^6b^{10}+864\cos(4\theta)a^4b^{12}-3456\cos(2\theta)a^{12}b^4-5376\cos(2\theta)a^{10}b^6+5376\cos(2\theta)a^6b^{10}+3456\cos(2\theta)a^4b^{12}+2592a^{12}b^4+7296a^{10}b^6-15680a^8b^8+7296a^6b^{10}+2592a^4b^{12})L^2+384\cos(6\theta)a^{12}b^6-1152a^{10}b^8\cos(6\theta)+1152a^8b^{10}\cos(6\theta)-384a^6b^{12}\cos(6\theta)-2304\cos(4\theta)a^{12}b^6+2304a^{10}b^8\cos(4\theta)+2304a^8b^{10}\cos(4\theta)-2304a^6b^{12}\cos(4\theta)+5760a^{12}b^6\cos(2\theta)+1152a^{10}b^8\cos(2\theta)-1152a^8b^{10}\cos(2\theta)-5760a^6b^{12}\cos(2\theta)-3840a^{12}b^6-2304a^{10}b^8-2304a^8b^{10}-3840a^6b^{12}=0\)



\((9L^2a^4+6L^2a^2b^2+L^2b^4-48a^4b^2)(3L^2a^4+6L^2a^2b^2-9L^2b^4-16a^4b^2)^2+48b^2(a-b)(a+b)(27L^4a^8+36L^4a^6b^2+130L^4a^4b^4+36L^4a^2b^6+27L^4b^8-288L^2a^8b^2-192L^2a^6b^4-544L^2a^4b^6+768a^8b^4)x^2+768b^4(a-b)^2(a+b)^2(9L^2a^4-2L^2a^2b^2+9L^2b^4-48a^4b^2)x^4+12288b^6(a-b)^3(a+b)^3x^6=0\)



另外，我们也能得到正三角形中心轨迹椭圆上的一点\([x_0,y_0]\)与三角形边长L的简单关系：

\(L^2a^4+6L^2a^2b^2+9L^2b^4-48a^4y_0^2-48a^2b^4+48a^2b^2y_0^2=0\)

\(9L^2a^4+6L^2a^2b^2+L^2b^4-48a^4b^2+48a^2b^2x_0^2-48b^4x_0^2=0\)