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青青子衿

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tT
发表于 2020-2-29 16:10 | 只看该作者

三条角平分线有理且整面积

本帖最后由 青青子衿 于 2020-2-29 17:51 编辑

三边长均不超过1000单位长度，三条角平分线有理且整面积
\begin{align*}
(\,\phantom{0}14,\enspace\phantom{0}25,\enspace\phantom{0}25\,)&&(\,231,\enspace250,\enspace289\,)\\
(\,\phantom{0}77,\enspace289,\enspace338\,)&&(\,286,\enspace841,\enspace875\,)\\
(\,\phantom{0}82,\enspace841,\enspace841\,)&&(\,289,\enspace289,\enspace322\,)\\
(\,\phantom{0}84,\enspace125,\enspace169\,)&&(\,289,\enspace338,\enspace399\,)\\
(\,\phantom{0}91,\enspace250,\enspace289\,)&&(\,325,\enspace578,\enspace825\,)\\
\\
(\,125,\enspace154,\enspace169\,)&&(\,350,\enspace429,\enspace625\,)\\
(\,125,\enspace507,\enspace578\,)&&\\
(\,150,\enspace169,\enspace275\,)&&\\
(\,169,\enspace169,\enspace238\,)&&\\
(\,204,\enspace841,\enspace875\,)&&\\
\end{align*}
...
  1. AbsoluteTiming[nn = 1000; lst = {};
  2. Do[s = (a + b + c)/2;
  3.   If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c);
  4.    l1 = b*c*(b + c + a) (b + c - a)/(b + c)^2;
  5.    l2 = a*c*(a + c + b) (a + c - b)/(a + c)^2;
  6.    l3 = a*b*(a + b + c) (a + b - c)/(a + b)^2;
  7.    If[0 < area2 &&
  8.      IntegerQ[
  9.       Sqrt[area2]] && (Element[Sqrt[l1], Rationals] &&
  10.        Element[Sqrt[l2], Rationals] && Element[Sqrt[l3], Rationals]) &&
  11.       GCD[a, b, c] == 1, AppendTo[lst, {a, b, c}]]], {a, nn}, {b,
  12.    a}, {c, b}];
  13. Grid[Transpose[
  14.    Partition[
  15.     Multicolumn[#, 1] & /@
  16.      Partition[SortBy[Sort /@ Union[lst], # &], 5, 5, {1, 1},
  17.       Style[{}, White]], 2]], Spacings -> {2, 2}]]
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青青子衿

2#
发表于 2020-3-10 17:10 | 只看该作者
本帖最后由 青青子衿 于 2020-3-10 19:08 编辑

回复 1# 青青子衿
\begin{align*}
(\,\phantom{00}14,\enspace\phantom{00}25,\enspace\phantom{00}25\,)&&(\,\phantom{0}867,\enspace1036,\enspace1681\,)&&(\,1375,\enspace2197,\enspace3468\,)\\
(\,\phantom{00}84,\enspace\phantom{0}125,\enspace\phantom{0}169\,)&&(\,\phantom{0}750,\enspace1309,\enspace1681\,)&&(\,1258,\enspace2809,\enspace3549\,)\\
(\,\phantom{0}125,\enspace\phantom{0}154,\enspace\phantom{0}169\,)&&(\,\phantom{0}750,\enspace1681,\enspace1729\,)&&(\,2873,\enspace3362,\enspace3625\,)\\
(\,\phantom{0}169,\enspace\phantom{0}169,\enspace\phantom{0}238\,)&&(\,1183,\enspace1369,\enspace1914\,)&&(\,1859,\enspace2170,\enspace3721\,)\\
(\,\phantom{0}150,\enspace\phantom{0}169,\enspace\phantom{0}275\,)&&(\,\phantom{0}867,\enspace1681,\enspace2002\,)&&(\,1375,\enspace3094,\enspace3721\,)\\
\\
(\,\phantom{00}91,\enspace\phantom{0}250,\enspace\phantom{0}289\,)&&(\,1045,\enspace1682,\enspace2023\,)&&(\,2197,\enspace3469,\enspace3721\,)\\
(\,\phantom{0}231,\enspace\phantom{0}250,\enspace\phantom{0}289\,)&&(\,1209,\enspace1682,\enspace2023\,)&&(\,2809,\enspace3549,\enspace3740\,)\\
(\,\phantom{0}289,\enspace\phantom{0}289,\enspace\phantom{0}322\,)&&(\,1014,\enspace1859,\enspace2125\,)&&(\,2023,\enspace3125,\enspace3762\,)\\
(\,\phantom{00}77,\enspace\phantom{0}289,\enspace\phantom{0}338\,)&&(\,1369,\enspace1369,\enspace2162\,)&&(\,1375,\enspace3721,\enspace3864\,)\\
(\,\phantom{0}289,\enspace\phantom{0}338,\enspace\phantom{0}399\,)&&(\,1014,\enspace1681,\enspace2233\,)&&(\,2002,\enspace2809,\enspace4107\,)\\
\\
(\,\phantom{0}125,\enspace\phantom{0}507,\enspace\phantom{0}578\,)&&(\,1156,\enspace1859,\enspace2375\,)&&(\,\phantom{0}551,\enspace3750,\enspace4225\,)\\
(\,\phantom{0}350,\enspace\phantom{0}429,\enspace\phantom{0}625\,)&&(\,1035,\enspace2023,\enspace2738\,)&&(\,\phantom{0}969,\enspace3850,\enspace4225\,)\\
(\,\phantom{0}325,\enspace\phantom{0}578,\enspace\phantom{0}825\,)&&(\,1118,\enspace2523,\enspace2809\,)&&(\,4094,\enspace4225,\enspace4225\,)\\
(\,\phantom{00}82,\enspace\phantom{0}841,\enspace\phantom{0}841\,)&&(\,1364,\enspace2523,\enspace2809\,)&&(\,2366,\enspace3125,\enspace4301\,)\\
(\,\phantom{0}204,\enspace\phantom{0}841,\enspace\phantom{0}875\,)&&(\,\phantom{0}506,\enspace2625,\enspace2809\,)&&(\,2100,\enspace4225,\enspace4301\,)\\
\\
(\,\phantom{0}286,\enspace\phantom{0}841,\enspace\phantom{0}875\,)&&(\,1976,\enspace2625,\enspace2809\,)&&(\,2809,\enspace4107,\enspace4484\,)\\
(\,\phantom{0}625,\enspace\phantom{0}625,\enspace1054\,)&&(\,2482,\enspace2809,\enspace2809\,)&&(\,1859,\enspace3721,\enspace4788\,)\\
(\,\phantom{0}792,\enspace\phantom{0}841,\enspace1183\,)&&(\,1681,\enspace1681,\enspace3038\,)&&(\,2738,\enspace3125,\enspace4797\,)\\
(\,\phantom{0}841,\enspace\phantom{0}874,\enspace1183\,)&&(\,1508,\enspace2023,\enspace3125\,)&&\\
(\,\phantom{0}836,\enspace\phantom{0}875,\enspace1369\,)&&(\,\phantom{0}969,\enspace2366,\enspace3125\,)&&\\
\\
(\,\phantom{0}841,\enspace1040,\enspace1369\,)&&(\,1682,\enspace2553,\enspace3125\,)&&\\
(\,\phantom{0}841,\enspace1122,\enspace1369\,)&&(\,1682,\enspace2717,\enspace3125\,)&&\\
(\,\phantom{0}248,\enspace1183,\enspace1369\,)&&(\,\phantom{0}473,\enspace2738,\enspace3125\,)&&\\
(\,\phantom{0}875,\enspace1326,\enspace1369\,)&&(\,2023,\enspace2738,\enspace3289\,)&&\\
(\,\phantom{0}805,\enspace1014,\enspace1681\,)&&(\,1250,\enspace2523,\enspace3325\,)&&\\
\end{align*}

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