整边三角形

ccmmjj

两个三角形，有一个角是相同的，它们的边长，是六个不同的一位整数，求这两个三角形。

wayne

暴力搜索：

1. tmp=Flatten[Table[Select[Permutations[t],#[[1]]<#[[4]]&&#[[1]]<#[[2]]&&#[[4]]<#[[5]]&&(#[[1]]^2+#[[2]]^2-#[[3]]^2)/(#[[1]]*#[[2]])==(#[[4]]^2+#[[5]]^2-#[[6]]^2)/(#[[4]]*#[[5]])&],{t,Subsets[Range[9],{6}]}],1];Select[tmp,#[[1]]+#[[2]]>#[[3]]&&Abs[#[[1]]-#[[2]]]<#[[3]]&]

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{{2, 4, 3, 7, 8, 6}}

2, 4, 3 和 7, 8, 6

wayne

再放大范围，边长小于20的，且不是相似三角形的，有：

1. {{2,3,4,7,6,8},{3,8,10,5,4,6},{2,4,5,9,8,10},{2,5,6,4,7,10},{3,10,11,4,5,7},{2,5,6,11,10,12},{2,3,4,8,5,12},{2,11,12,4,5,6},{2,8,9,3,10,12},{5,12,13,6,8,10},{7,8,10,13,12,14},{7,8,12,10,9,14},{7,11,12,9,13,14},{6,12,14,7,8,9},{6,10,14,7,8,13},{6,10,14,7,5,8},{5,6,7,10,8,14},{4,5,7,12,10,14},{4,11,14,10,7,12},{4,11,12,8,13,14},{4,6,7,8,11,14},{4,12,14,7,8,10},{4,7,9,6,12,14},{3,6,8,12,9,14},{3,13,14,7,11,12},{3,7,8,6,10,14},{2,6,7,13,12,14},{2,5,6,10,11,14},{2,13,14,7,10,12},{8,9,15,10,6,12},{7,12,15,9,11,14},{6,8,10,14,13,15},{6,12,15,9,8,10},{5,6,7,11,14,15},{4,5,7,14,11,15},{4,6,9,12,8,15},{4,13,15,6,8,10},{4,9,10,6,14,15},{3,4,5,14,13,15},{3,5,6,12,13,15},{3,5,6,9,8,15},{3,9,10,5,12,15},{2,6,7,4,14,15},{9,11,16,10,8,12},{8,9,10,13,15,16},{8,10,12,11,15,16},{8,13,15,10,14,16},{7,11,16,12,8,14},{7,13,15,10,14,16},{7,12,13,10,14,16},{7,8,12,9,14,16},{6,14,16,8,13,15},{6,14,16,7,13,15},{6,14,16,7,10,13},{5,15,16,10,12,11},{5,6,8,10,13,16},{5,13,16,8,11,15},{5,14,16,6,9,12},{5,14,16,6,7,8},{5,7,8,6,14,16},{4,8,10,15,13,16},{4,6,8,14,12,16},{4,5,6,11,15,16},{4,5,6,11,9,16},{4,5,8,10,9,16},{4,12,14,7,13,16},{4,10,12,5,15,16},{4,7,10,5,15,16},{4,6,8,5,14,16},{3,8,10,15,11,16},{3,5,7,14,10,16},{3,7,8,10,14,16},{3,8,10,9,11,16},{3,14,16,8,7,10},{3,8,10,6,11,16},{3,5,7,6,14,16},{3,8,10,4,15,16},{2,7,8,15,14,16},{2,4,5,15,13,16},{2,3,4,14,12,16},{2,3,4,5,14,16},{2,9,10,3,15,16},{6,7,8,11,16,17},{5,11,12,15,17,16},{5,12,13,8,15,17},{4,15,17,5,8,11},{4,14,17,5,8,12},{3,4,5,8,15,17},{3,16,17,6,9,12},{3,16,17,4,6,8},{10,17,18,12,14,13},{10,14,18,9,15,16},{9,11,12,15,17,18},{9,11,16,15,12,18},{9,10,13,11,15,18},{8,13,14,16,18,17},{8,11,18,15,6,16},{8,13,18,6,14,15},{7,9,12,16,14,18},{7,13,18,14,9,15},{7,17,18,9,11,10},{7,9,12,8,14,18},{6,9,10,14,17,18},{6,10,11,12,15,18},{6,15,18,11,10,12},{6,10,12,11,15,18},{6,15,18,10,11,14},{6,13,14,10,17,18},{6,11,15,10,12,18},{6,16,18,9,13,17},{6,16,18,9,11,12},{6,16,18,8,9,13},{6,9,10,8,16,18},{5,9,10,11,15,18},{5,8,9,10,14,18},{5,7,9,10,16,18},{5,17,18,9,11,10},{5,9,12,7,15,18},{5,9,11,6,16,18},{4,11,12,16,18,17},{4,7,9,16,14,18},{4,8,9,14,17,18},{4,14,15,10,17,18},{4,9,11,10,12,18},{4,9,10,8,13,18},{4,7,10,6,15,18},{3,8,9,15,17,18},{3,8,10,15,12,18},{3,6,7,14,16,18},{3,5,6,11,15,18},{3,17,18,9,14,15},{3,6,7,8,14,18},{3,4,6,8,16,18},{3,10,12,4,16,18},{2,8,9,17,16,18},{9,14,19,10,12,18},{8,12,13,14,16,19},{7,12,13,14,15,19},{7,10,13,11,14,19},{6,7,8,15,16,19},{6,9,12,7,16,19},{5,7,8,14,15,19},{5,16,19,7,8,13},{5,18,19,6,10,12},{5,16,19,6,10,14},{4,10,11,12,14,19},{4,8,9,10,16,19},{4,6,8,7,16,19},{3,7,8,11,14,19},{2,3,4,7,16,19},{2,18,19,6,8,12},{2,18,19,3,4,6}}

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ccmmjj

谢谢，会使用软件就是好。

倪举鹏

边长是整数 面积是整数

xiaoshuchong

倪举鹏 发表于 2015-5-7 19:33
边长是整数 面积是整数

对于n>m且n,m均为整数，如下三角形面积为整数且其中一角为$\arccos(3/5)$
\[\begin{eqnarray*}
a,b,c&=&-5m^{2}+5n^{2},10mn+6n^{2},5m^{2}+6mn+5n^{2}\\S&=&4n\left(n^{2}-m^{2}\right)\left(5m+3n\right)
\end{eqnarray*}\]
n小于等于10有如下例子
[15, 44, 37]
[10, 21, 17]
[75, 136, 109]
[3, 5, 4]
[175, 276, 221]
[60, 91, 73]
[315, 464, 373]
[25, 36, 29]
[99, 140, 113]
[25, 114, 101]
[21, 50, 41]
[225, 434, 349]
[385, 666, 533]
[35, 216, 197]
[4, 15, 13]
[25, 63, 52]
[275, 624, 509]
[91, 180, 145]
[9, 70, 65]
[165, 574, 493]
[325, 846, 701]
[55, 516, 485]
[30, 161, 145]
[195, 784, 685]
[35, 117, 100]
[65, 714, 677]
[75, 944, 901]
[40, 279, 257]
[51, 260, 233]
[85, 1206, 1157]
[19, 300, 289]

类似地，还可以构造其中一角为$\arccos(a/c)$ (a,b,c为勾股数)的三角形。

ejsoon

邊長2,4,3的三角形有神奇的自相似組合。