三角形面积公式

王守恩

三角形3边为 \(\sqrt{a},\sqrt{b},\sqrt{c}\),   三角形面积=\(\frac{\sqrt{4ab - (a + b - c)^2\ }}{4}\)

三角形3边为 \(a,b,c\),   三角形面积=\(\frac{\sqrt{4a^2b^2 - (a^2 + b^2 - c^2)^2\ \ }}{4}\)

王守恩

三角形三边长为√a,√b,√c,  面积=36, 这样的三角形有几个？
答案太多了，加几条：a,b,c是整数。  a,b,c两两互素。

王守恩

三角形三边长为√a,√b,√c, 面积=36, 这样的三角形有几个？
答案太多了，加几条：a,b,c是整数。 a,b,c两两互素。
答案还是多，再加：a + b > c > b > a > c - b > 0
先从简单的数开始。
边长为√a,√b,√c, 面积=n, 这样的三角形有几个？
n=03,a(03)=01,{5,8,9},
n=04,a(04)=01,{5,13,16},
n=05,a(05)=01,{8,13,17},
n=06,a(06)=02,{5,29,32}, {9,17,20},
n=07,a(07)=02,{8,25,29}, {13,17,20},
n=08,a(08)=03,{5,52,53}, {13,20,29}, {16,17,25},
n=09,a(09)=04,{8,41,45}, {9,37,40}, {13,25,36}, {17,20,29},
n=10,a(10)=02,{13,32,37}, {17,25,32},
n=11,a(11)=03,{8,61,65}, {13,40,41}, {17,29,40},
n=12,a(12)=04,{5,116,117}, {9,65,68}, {16,37,45}, {17,36,41},
n=13,a(13)=04,{5,136,137}, {8,85,89}, {17,40,53}, {20,37,41}
n=14,a(14)=05,{13,61,68}, {17,49,52}, {20,41,49}, {25,32,49}, {29,32,37},
n=15,a(15)=05,{8,113,117}, {9,101,104}, {13,72,73}, {25,37,52}, {29,36,41},
n=16,a(16)=05,{16,65,73}, {17,64,65}, {20,53,61}, {25,41,64}, {29,37,52},
n=17,a(17)=05,{5,232,233}, {8,145,149}, {13,89,100}, {25,52,53}, {37,40,41},
n=18,a(18)=08,{9,145,148}, {13,100,109}, {17,80,81}, {25,52,73}, {29,45,68}, {32,41,65}, {32,45,53}, {36,37,61}
n=19,a(19)=05,{5,289,292}, {8,181,185}, {13,113,116}, {25,61,68}, {40,41,53},
n=20,a(20)=06,{13,125,128}, {16,101,109}, {17,97,100}, {25,68,73}, {29,61,64}, {37,52,53},
n=21,a(21)=11,{5,353,356}, {8,221,225}, {9,197,200}, {13,136,145}, {20,89,101}, {29,65,72}, {36,53,65}, {37,49,72}, {40,49,61}, {41,45,68}, {45,49,52},
n=22,a(22)=07,{5,388,389}, {13,149,160}, {17,116,121}, {20,97,113}, {32,61,85}, {32,65,73}, {37,53,80},
n=23,a(23)=06,{8,265,269}, {20,109,113}, {29,73,100}, {37,65,68}, {40,53,89}, {41,52,85},
n=24,a(24)=10,{5,461,464}, {9,257,260}, {13,180,181}, {16,145,153}, {17,137,144}, {29,80,101}, {36,65,89}, {37,64,85}, {41,65,68}, {45,61,64},
n=25,a(25)=09,{8,313,317}, {13,193,200}, {17,148,157}, {25,101,116}, {25,104,109}, {29,89,100}, {37,68,97}, {41,61,100}, {52,53,73}

nyy

王守恩 发表于 2024-7-1 16:10
三角形三边长为√a,√b,√c, 面积=36, 这样的三角形有几个？
答案太多了，加几条：a,b,c是整数。 a,b,c两两 ...

你是用穷举法吗？

王守恩

nyy 发表于 2024-7-2 08:27
你是用穷举法吗？

我只会用这个蹩脚的公式。3#是我一个一个一个一个删出来的。可费劲了。

1. Table[NSolve[{k == Sqrt[4 a b - (a + b - c)^2]/4, GCD[a, b] == GCD[b, c] == GCD[c, a] == 1,  a + b > c > b > a > c - b > 0}, {a, b, c}, Integers], {k, 1, 13}]

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{{}, {}, {{a -> 5, b -> 8, c -> 9}}, {{a -> 5, b -> 13, c -> 16}}, {{a -> 8, b -> 13, c -> 17}}, {{a -> 5, b -> 29, c -> 32}, {a -> 9, b -> 17, c -> 20}},
{{a -> 8, b -> 25, c -> 29}, {a -> 13, b -> 17, c -> 20}}, {{a -> 5, b -> 52,  c -> 53}, {a -> 13, b -> 20, c -> 29}, {a -> 16, b -> 17,  c -> 25}},
{{a -> 8, b -> 41, c -> 45}, {a -> 9, b -> 37,  c -> 40}, {a -> 13, b -> 25, c -> 36}, {a -> 17, b -> 20,  c -> 29}}, {{a -> 13, b -> 32, c -> 37}, {a -> 17, b -> 25, c -> 32}},
{{a -> 8, b -> 61, c -> 65}, {a -> 13, b -> 40, c -> 41}, {a -> 17, b -> 29, c -> 40}}, {{a -> 5, b -> 116, c -> 117}, {a -> 9, b -> 65, c -> 68}, {a -> 16, b -> 37,  c -> 45},
{a -> 17, b -> 36, c -> 41}}, {{a -> 5, b -> 136,  c -> 137}, {a -> 8, b -> 85, c -> 89}, {a -> 17, b -> 40,  c -> 53}, {a -> 20, b -> 37, c -> 41}}}

northwolves

王守恩 发表于 2024-7-2 09:33
我只会用这个蹩脚的公式。3#是我一个一个一个一个删出来的。可费劲了。

{{}, {}, {{a -> 5, b -> 8, c - ...

为什么要一个一个一个一个删除呢？

1. Table[s =
   
2.    Values@NSolve[{16 k^2 == 4 a b - (a + b - c)^2,
   
3.       GCD[a, b] == GCD[b, c] == GCD[c, a] == 1,
   
4.       a + b > c > b > a > c - b}, {a, b, c}, Integers]; {k, Length@s,
   
5.    s}, {k, 1, 8}] // MatrixForm

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1        0        {}
2        0        {}
3        1        {{5,8,9}}
4        1        {{5,13,16}}
5        1        {{8,13,17}}
6        2        {{5,29,32},{9,17,20}}
7        2        {{8,25,29},{13,17,20}}
8        3        {{5,52,53},{13,20,29},{16,17,25}}

王守恩

northwolves 发表于 2024-7-2 11:58
为什么要一个一个一个一个删除呢？

a(03)=9,
a(04)=16,
a(05)=17,
a(06)=20,
a(07)=20,
a(08)=25,
a(09)=29,
a(10)=32,
a(11)=40,
a(12)=41,
......

得到这样一串数: 9, 16, 17, 20, 20, 25, 29, 32, 40, 41, 41, 37, 41, 52, 41, 61, 53, 53, 52, 80, 85, 64, 73, ... 还可以拉出来吗？谢谢！
  

northwolves

1. f[k_]:=Module[{s=Flatten@Table[Values@Solve[{16 k^2==4 a b-(a+b-c)^2,GCD[a,b]==GCD[b,c]==GCD[c,a]==1,a+b>c>b>a>c-b},{b,c},Integers],{a,1,(7k+1)/3}]},s[[-1]]]
   
2. Table[f[k], {k, 3, 10}]
   

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{9, 16, 17, 20, 20, 25, 29, 32}

northwolves

1. f[k_]:=For[a=Floor[(7k+1)/3],a>0,a--,s=Values@Solve[{16 k^2==4 a b-(a+b-c)^2,GCD[a,b]==GCD[b,c]==GCD[c,a]==1,a+b>c>b>a>c-b},{b,c},Integers];If [Length@s>0,Return[Flatten[s][[-1]]]]]
   
2. 
   
3. Table[f[k], {k, 3, 20}]

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{9, 16, 17, 20, 20, 25, 29, 32, 40, 41, 41, 37, 41, 52, 41, 61, 53, 53}

王守恩

northwolves 发表于 2024-7-2 13:59
{9, 16, 17, 20, 20, 25, 29, 32, 40, 41, 41, 37, 41, 52, 41, 61, 53, 53}

找不到规律！丢了。换一道！

a(03)=1, {1,1,1},
a(04)=0,
a(05)=1, {1,2,2},
a(06)=1, {2,2,2},
a(07)=2, {1,3,3},{2,2,3},
a(08)=1, {2,3,3}
a(09)=3, {1,4,4},{2,3,4},{3,3,3},
a(10)=2, {2,4,4},{3,3,4},
a(11)=4, {1,5,5},{2,4,5},{3,3,5},{3,4,4},
a(12)=3, {2,5,5},{3,4,5},{4,4,4},
a(13)=5, {1,6,6},{2,5,6},{3,4,6},{3,5,5},{4,4,5},
a(14)=4, {2,6,6},{3,5,6},{4,4,6},{4,5,5},
a(15)=7, {1,7,7},{2,6,7},{3,5,7},{3,6,6},{4,4,7},{4,5,6},{5,5,5},
a(16)=5, {2,7,7},{3,6,7},{4,5,7},{4,6,6},{5,5,6},
a(17)=8, {1,8,8},{2,7,8},{3,6,8},{3,7,7},{4,5,8},{4,6,7},{5,5,7},{5,6,6},
a(18)=6, {2,8,8},{3,7,8},{4,6,8},{5,5,8},{5,6,7}{6,6,6},
a(19)=0, {1,9,9},{2,8,9},{3,7,9},{3,8,8},{4,6,9},{4,7,8},{5,5,9},{5,6,8},{5,7,7},{6,6,7},

每个{ }都是三角形。大数不大于另2数和。