abcxyz对应的四面体体积公式大于零，那么abcxyz能构成四面体吗？

nyy · 发表于 2023-3-27 13:59:03

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abc是从同一个顶点出发的三条棱，xyz是abc的三条对棱。
假设abcxyz都是大于零的六个变量。那么当对应的体积公式对应的结果大于零的时候，
abcxyz能构成四面体吗？

这个问题源于三角形的面积公式海伦公式s^2=p(p-a)(p-b)(p-c)
限制abc都大于零，
不妨假设0<a<=b<=c
p(p-a)(p-b)(p-c)>0，则p的变量范围是(-无穷,0)∪(a,b)∪(c,正无穷)
由于abc都大于零，所以p的范围是(a,b)∪(c,正无穷)，
先讨论
a<(a+b+c)/2<b，左边的小于号明显是成立的，右边的小于号明显是不成立的（与b<=c矛盾），
因此得到0<a<=b<=c<p，因此得到任意两边的和都大于第三边，因此abc能构成一个三角形。

现在把问题推向四面体，试问：如果对应的四面体体积公式的值大于零的时候，能得到abcxyz能构成四面体吗？

下面这个是四面体体积公式的mathematica代码

fun[a_,b_,c_,x_,y_,z_]:=Sqrt[Det[{{0,1,1,1,1},{1,0,a^2,b^2,c^2},{1,a^2,0,z^2,y^2},{1,b^2,z^2,0,x^2},{1,c^2,y^2,x^2,0}}]/288]

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nyy · 发表于 2023-3-27 16:02:41

如果不能构成，能给出一个反例吗？

nyy · 发表于 2023-3-28 14:29:02

nyy 发表于 2023-3-27 16:02
如果不能构成，能给出一个反例吗？

反例很多，我自己就能穷举法很多出来
{a,b,c,x,y,z}结果如下：

{25,61,97,13,68,32}
{17,53,78,20,98,75}
{82,14,95,53,10,32}
{67,94,25,62,31,14}
{16,68,97,24,62,10}
{72,23,24,52,21,97}
{10,51,25,91,43,24}
{32,16,68,96,17,62}
{93,19,47,23,20,59}
{11,46,10,93,48,11}
{87,62,12,10,46,22}
{41,18,25,84,10,96}
{19,65,14,82,44,23}
{13,33,87,11,53,16}
{20,12,67,90,26,52}

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貌似反例非常的多！

nyy · 发表于 2023-3-28 14:35:29

nyy 发表于 2023-3-28 14:29
反例很多，我自己就能穷举法很多出来
{a,b,c,x,y,z}结果如下：

Clear["Global`*"];(*删除所有变量*)
(*子函数,四面体体积公式，a,b,c分别是从一个顶点出发的三条棱，x,y,z分别是对棱*)
fun[a_,b_,c_,x_,y_,z_]:=Sqrt[Det[{{0,1,1,1,1},{1,0,a^2,b^2,c^2},{1,a^2,0,z^2,y^2},{1,b^2,z^2,0,x^2},{1,c^2,y^2,x^2,0}}]/288]
(*子函数,海伦公式*)
heron[a_,b_,c_]:=Module[{p=(a+b+c)/2},Sqrt[p*(p-a)*(p-b)*(p-c)]]
Do[
(*随机产生六条棱*)
a=RandomInteger[{10,99}];
b=RandomInteger[{10,99}];
c=RandomInteger[{10,99}];
x=RandomInteger[{10,99}];
y=RandomInteger[{10,99}];
z=RandomInteger[{10,99}];
vol=fun[a,b,c,x,y,z]^2;(*计算体积的平方*)
(*用海伦公式计算4个面的面积的平方,如果能构成三角形,则面积平方大于零,否则小于零*)
s1=heron[a,b,z]^2;
s2=heron[a,c,y]^2;
s3=heron[x,y,z]^2;
s4=heron[b,c,x]^2;
(*如果体积的平方大于零,并且其中一个面不能构成三角形,则输出结果*)
If[And[vol>0,Or[s1<0,s2<0,s3<0,s4<0]],Print[{a,b,c,x,y,z}]]
,{k,3000}(*重复3000次*)
]

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给出我的代码，如果试验个100次，还是能找到结果的！虽然看样子满足要求的结果还是比较少的！

nyy · 发表于 2023-3-28 14:51:02

本帖最后由 nyy 于 2023-3-28 14:55 编辑

nyy 发表于 2023-3-28 14:29
反例很多，我自己就能穷举法很多出来
{a,b,c,x,y,z}结果如下：

给出六条棱，并且给出四个面的面积符号（能构成三角形则+1，否则-1），输出结果如下：

{99,15,49,13,12,73,-1,-1,-1,-1}
{10,58,10,13,46,77,-1,-1,-1,-1}
{11,39,47,99,13,72,-1,-1,-1,-1}
{53,12,10,32,39,89,-1,-1,-1,-1}
{12,49,95,16,53,18,-1,-1,-1,-1}
{86,10,68,32,12,45,-1,-1,-1,-1}
{12,42,19,16,68,89,-1,-1,-1,-1}
{24,87,47,13,20,56,-1,-1,-1,-1}
{45,14,13,83,14,98,-1,-1,-1,-1}
{40,97,64,17,13,56,-1,-1,-1,-1}
{60,12,28,51,12,89,-1,-1,-1,-1}
{92,20,55,16,17,61,-1,-1,-1,-1}
{23,18,66,22,91,49,-1,-1,-1,-1}
{54,89,15,53,19,20,-1,-1,-1,-1}
{65,96,23,61,10,24,-1,-1,-1,-1}
{21,63,95,23,61,24,-1,-1,-1,-1}
{38,11,38,55,93,14,-1,-1,-1,-1}
{46,95,10,73,22,32,-1,-1,-1,-1}
{39,26,46,15,99,74,-1,-1,-1,-1}
{95,50,22,22,49,14,-1,-1,-1,-1}
{39,10,84,73,12,27,-1,-1,-1,-1}
{93,26,46,11,26,47,-1,-1,-1,-1}
{95,12,69,41,23,72,-1,-1,-1,-1}
{35,64,28,99,69,25,-1,-1,-1,-1}
{94,56,27,14,59,17,-1,-1,-1,-1}
{12,56,11,88,42,25,-1,-1,-1,-1}

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从上面可以看出：如果四面体体积大于零，
并且abcxyz不构成四面体，则四个面都不构成三角形，
问题来了：如果体积表达式大于零，并且其中一个面也构成三角形，
此时是否就能构成四面体了？

Clear["Global`*"];(*删除所有变量*)
(*子函数,四面体体积公式，a,b,c分别是从一个顶点出发的三条棱，x,y,z分别是对棱*)
fun[a_,b_,c_,x_,y_,z_]:=Sqrt[Det[{{0,1,1,1,1},{1,0,a^2,b^2,c^2},{1,a^2,0,z^2,y^2},{1,b^2,z^2,0,x^2},{1,c^2,y^2,x^2,0}}]/288]
(*子函数,海伦公式*)
heron[a_,b_,c_]:=Module[{p=(a+b+c)/2},Sqrt[p*(p-a)*(p-b)*(p-c)]]
Do[
(*随机产生六条棱*)
a=RandomInteger[{10,99}];
b=RandomInteger[{10,99}];
c=RandomInteger[{10,99}];
x=RandomInteger[{10,99}];
y=RandomInteger[{10,99}];
z=RandomInteger[{10,99}];
vol=fun[a,b,c,x,y,z]^2;(*计算体积的平方*)
(*用海伦公式计算4个面的面积的平方,如果能构成三角形,则面积平方大于零,否则小于零*)
s1=heron[a,b,z]^2;If[s1>0,k1="+1",k1="-1"];(*k1是面积符号*)
s2=heron[a,c,y]^2;If[s2>0,k2="+1",k2="-1"];
s3=heron[x,y,z]^2;If[s3>0,k3="+1",k3="-1"];
s4=heron[b,c,x]^2;If[s4>0,k4="+1",k4="-1"];
(*如果体积的平方大于零,并且其中一个面不能构成三角形,则输出结果*)
If[And[vol>0,Or[s1<0,s2<0,s3<0,s4<0]],Print[{a,b,c,x,y,z,k1,k2,k3,k4}]]
,{k,3000}(*重复3000次*)
]

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nyy · 发表于 2023-3-29 08:55:14

我个人怀疑：如果四面体的体积表达式大于零，那么要么四个面都能构成三角形，要么四个面都不能构成三角形！
不存在有的面能构成三角形且有的面不能构成三角形的情况。

全部代码如下：

Clear["Global`*"];(*删除所有变量*)
(*子函数,四面体体积公式，a,b,c分别是从一个顶点出发的三条棱，x,y,z分别是对棱*)
fun[a_,b_,c_,x_,y_,z_]:=Sqrt[Det[{{0,1,1,1,1},{1,0,a^2,b^2,c^2},{1,a^2,0,z^2,y^2},{1,b^2,z^2,0,x^2},{1,c^2,y^2,x^2,0}}]/288]
(*子函数,海伦公式*)
heron[a_,b_,c_]:=Module[{p=(a+b+c)/2},Sqrt[p*(p-a)*(p-b)*(p-c)]]
Do[
(*随机产生六条棱*)
a=RandomInteger[{10,99}];
b=RandomInteger[{10,99}];
c=RandomInteger[{10,99}];
x=RandomInteger[{10,99}];
y=RandomInteger[{10,99}];
z=RandomInteger[{10,99}];
vol=fun[a,b,c,x,y,z]^2;(*计算体积的平方*)
(*用海伦公式计算4个面的面积的平方,如果能构成三角形,则面积平方大于零,否则小于零*)
s1=heron[a,b,z]^2;If[s1>0,k1="+1",k1="-1"];(*k1是面积符号*)
s2=heron[a,c,y]^2;If[s2>0,k2="+1",k2="-1"];
s3=heron[x,y,z]^2;If[s3>0,k3="+1",k3="-1"];
s4=heron[b,c,x]^2;If[s4>0,k4="+1",k4="-1"];
(*如果体积的平方大于零,并且其中一个面不能构成三角形,则输出结果*)
If[And[vol>0,Or[1>0,s1>0,s2>0,s3>0,s4>0]],Print[{a,b,c,x,y,z,k1,k2,k3,k4}]]
,{k,3000}(*重复3000次*)
]

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数值证据如下：

{99,92,95,63,84,93,+1,+1,+1,+1}
{77,90,86,29,56,38,+1,+1,+1,+1}
{89,84,82,73,76,38,+1,+1,+1,+1}
{91,74,61,61,59,93,+1,+1,+1,+1}
{58,91,81,55,69,80,+1,+1,+1,+1}
{88,78,87,90,94,94,+1,+1,+1,+1}
{95,58,17,57,99,68,+1,+1,+1,+1}
{86,73,92,62,68,87,+1,+1,+1,+1}
{97,94,72,49,46,56,+1,+1,+1,+1}
{86,54,96,68,32,46,+1,+1,+1,+1}
{58,79,56,86,96,93,+1,+1,+1,+1}
{56,49,36,31,45,56,+1,+1,+1,+1}
{49,63,55,62,49,25,+1,+1,+1,+1}
{99,84,66,83,91,80,+1,+1,+1,+1}
{46,76,17,66,58,83,+1,+1,+1,+1}
{73,75,91,92,81,98,+1,+1,+1,+1}
{24,68,13,76,26,85,+1,+1,+1,+1}
{44,33,18,40,33,70,+1,+1,+1,+1}
{77,79,32,83,69,52,+1,+1,+1,+1}
{14,48,45,62,40,55,+1,+1,+1,+1}
{77,75,50,92,62,89,+1,+1,+1,+1}
{43,92,63,91,44,67,+1,+1,+1,+1}
{54,81,99,93,51,63,+1,+1,+1,+1}
{60,85,92,77,42,49,+1,+1,+1,+1}
{94,71,93,73,88,73,+1,+1,+1,+1}
{57,56,88,78,79,32,+1,+1,+1,+1}
{81,59,73,44,59,76,+1,+1,+1,+1}
{19,98,34,18,13,59,-1,-1,-1,-1}
{88,59,35,50,83,69,+1,+1,+1,+1}
{81,92,52,85,68,62,+1,+1,+1,+1}
{61,54,55,31,60,56,+1,+1,+1,+1}
{95,46,71,27,79,77,+1,+1,+1,+1}
{42,59,57,26,42,60,+1,+1,+1,+1}
{77,85,42,50,51,67,+1,+1,+1,+1}
{39,79,67,66,71,92,+1,+1,+1,+1}
{77,70,45,37,56,58,+1,+1,+1,+1}
{53,65,76,46,98,83,+1,+1,+1,+1}
{91,90,83,71,76,83,+1,+1,+1,+1}
{56,52,26,68,62,96,+1,+1,+1,+1}
{19,22,53,68,61,18,+1,+1,+1,+1}
{84,94,91,30,52,77,+1,+1,+1,+1}
{55,46,56,61,39,80,+1,+1,+1,+1}
{15,87,30,78,27,86,+1,+1,+1,+1}
{84,47,89,61,83,69,+1,+1,+1,+1}
{98,67,99,38,79,56,+1,+1,+1,+1}
{19,80,40,10,10,40,-1,-1,-1,-1}
{70,80,52,67,85,85,+1,+1,+1,+1}
{54,47,74,67,95,39,+1,+1,+1,+1}
{42,70,34,77,66,51,+1,+1,+1,+1}
{62,23,78,69,34,47,+1,+1,+1,+1}
{23,57,35,27,33,48,+1,+1,+1,+1}
{58,55,98,67,48,29,+1,+1,+1,+1}
{70,69,93,59,89,40,+1,+1,+1,+1}
{41,65,32,67,63,96,+1,+1,+1,+1}
{47,58,20,59,64,81,+1,+1,+1,+1}
{53,49,65,61,41,66,+1,+1,+1,+1}
{94,74,37,61,78,63,+1,+1,+1,+1}
{93,60,62,31,54,38,+1,+1,+1,+1}
{47,38,61,50,63,46,+1,+1,+1,+1}
{59,21,74,73,62,57,+1,+1,+1,+1}
{99,61,51,69,69,63,+1,+1,+1,+1}
{67,82,33,92,56,92,+1,+1,+1,+1}
{71,66,84,85,34,84,+1,+1,+1,+1}
{64,25,55,38,69,50,+1,+1,+1,+1}
{84,99,81,59,65,73,+1,+1,+1,+1}
{43,38,55,56,96,73,+1,+1,+1,+1}
{57,77,73,44,61,53,+1,+1,+1,+1}
{90,83,40,48,52,43,+1,+1,+1,+1}
{60,93,55,76,57,84,+1,+1,+1,+1}
{49,72,79,89,51,67,+1,+1,+1,+1}
{98,29,72,53,51,91,+1,+1,+1,+1}
{69,57,71,49,41,25,+1,+1,+1,+1}
{43,44,55,68,64,55,+1,+1,+1,+1}
{88,74,85,36,79,63,+1,+1,+1,+1}
{77,67,41,68,94,36,+1,+1,+1,+1}
{27,58,16,79,56,10,-1,-1,-1,-1}
{66,45,29,59,76,97,+1,+1,+1,+1}
{35,80,72,91,48,93,+1,+1,+1,+1}
{56,45,41,69,85,57,+1,+1,+1,+1}
{35,72,38,62,27,61,+1,+1,+1,+1}
{45,66,37,53,52,86,+1,+1,+1,+1}
{15,50,99,90,92,49,+1,+1,+1,+1}
{96,94,98,80,81,27,+1,+1,+1,+1}
{46,87,58,75,79,95,+1,+1,+1,+1}
{58,91,55,98,97,94,+1,+1,+1,+1}
{94,69,52,69,73,64,+1,+1,+1,+1}
{83,64,87,30,90,64,+1,+1,+1,+1}
{50,50,55,54,65,59,+1,+1,+1,+1}
{57,71,83,60,44,51,+1,+1,+1,+1}
{60,51,82,47,64,28,+1,+1,+1,+1}
{66,40,36,54,79,28,+1,+1,+1,+1}
{68,44,72,58,92,88,+1,+1,+1,+1}
{96,44,84,97,34,95,+1,+1,+1,+1}
{65,39,48,22,85,67,+1,+1,+1,+1}
{23,51,92,31,68,18,-1,-1,-1,-1}
{68,38,73,57,98,70,+1,+1,+1,+1}
{94,64,39,38,75,85,+1,+1,+1,+1}
{69,48,90,92,41,82,+1,+1,+1,+1}
{30,52,81,86,57,56,+1,+1,+1,+1}
{73,66,29,69,71,19,+1,+1,+1,+1}
{97,81,81,19,29,25,+1,+1,+1,+1}
{45,69,44,45,13,45,+1,+1,+1,+1}
{64,73,80,38,31,53,+1,+1,+1,+1}
{88,79,26,76,93,49,+1,+1,+1,+1}
{55,50,48,78,14,82,+1,+1,+1,+1}
{69,13,68,70,66,71,+1,+1,+1,+1}
{62,54,53,53,46,34,+1,+1,+1,+1}
{40,69,49,55,48,51,+1,+1,+1,+1}
{52,68,95,60,91,73,+1,+1,+1,+1}
{83,76,81,37,53,65,+1,+1,+1,+1}
{81,77,62,58,53,40,+1,+1,+1,+1}
{97,95,42,56,64,27,+1,+1,+1,+1}
{94,71,54,37,74,62,+1,+1,+1,+1}
{92,94,47,98,76,65,+1,+1,+1,+1}
{53,34,57,65,59,56,+1,+1,+1,+1}
{67,39,92,95,71,68,+1,+1,+1,+1}
{94,88,89,49,52,87,+1,+1,+1,+1}
{13,35,46,93,20,61,-1,-1,-1,-1}
{47,30,45,55,42,71,+1,+1,+1,+1}
{41,57,62,76,57,45,+1,+1,+1,+1}
{77,48,35,71,56,83,+1,+1,+1,+1}
{23,74,60,29,71,87,+1,+1,+1,+1}
{44,62,35,42,10,41,+1,+1,+1,+1}
{69,46,64,69,58,40,+1,+1,+1,+1}
{80,59,48,75,70,56,+1,+1,+1,+1}
{44,54,59,84,93,89,+1,+1,+1,+1}
{96,63,39,78,84,85,+1,+1,+1,+1}
{82,70,58,61,47,44,+1,+1,+1,+1}
{76,53,46,46,58,35,+1,+1,+1,+1}
{72,31,45,36,56,82,+1,+1,+1,+1}
{95,31,44,67,90,98,+1,+1,+1,+1}
{96,72,53,88,85,45,+1,+1,+1,+1}
{56,60,17,64,52,89,+1,+1,+1,+1}
{79,65,44,24,44,42,+1,+1,+1,+1}
{25,50,64,55,60,60,+1,+1,+1,+1}
{98,71,59,73,72,82,+1,+1,+1,+1}
{33,90,51,29,15,55,-1,-1,-1,-1}
{83,99,68,96,52,89,+1,+1,+1,+1}
{78,64,37,58,58,29,+1,+1,+1,+1}
{39,96,33,92,60,98,+1,+1,+1,+1}
{55,88,41,73,62,89,+1,+1,+1,+1}
{88,59,79,49,66,67,+1,+1,+1,+1}
{96,38,87,51,87,81,+1,+1,+1,+1}
{55,49,38,65,49,52,+1,+1,+1,+1}
{74,68,43,64,51,82,+1,+1,+1,+1}
{96,53,48,82,75,94,+1,+1,+1,+1}
{78,32,28,40,83,73,+1,+1,+1,+1}
{56,55,62,75,64,89,+1,+1,+1,+1}
{64,43,61,63,42,37,+1,+1,+1,+1}
{73,87,43,77,58,35,+1,+1,+1,+1}
{31,51,66,32,52,29,+1,+1,+1,+1}
{36,13,95,74,50,19,-1,-1,-1,-1}
{72,40,80,50,24,40,+1,+1,+1,+1}
{67,91,92,74,95,66,+1,+1,+1,+1}
{48,67,84,56,69,64,+1,+1,+1,+1}
{37,44,86,92,84,58,+1,+1,+1,+1}
{84,98,94,88,46,84,+1,+1,+1,+1}
{68,50,56,63,69,29,+1,+1,+1,+1}
{80,72,48,87,86,93,+1,+1,+1,+1}
{45,66,59,47,91,99,+1,+1,+1,+1}
{68,60,53,61,77,77,+1,+1,+1,+1}
{82,18,69,76,36,92,+1,+1,+1,+1}
{78,64,55,86,46,94,+1,+1,+1,+1}
{71,95,88,94,98,55,+1,+1,+1,+1}
{81,21,52,25,10,46,-1,-1,-1,-1}
{62,55,81,83,64,71,+1,+1,+1,+1}
{77,56,72,50,57,71,+1,+1,+1,+1}
{47,61,49,33,46,67,+1,+1,+1,+1}
{37,70,75,99,48,58,+1,+1,+1,+1}
{62,89,58,72,48,69,+1,+1,+1,+1}
{90,83,94,66,75,50,+1,+1,+1,+1}
{44,37,53,20,46,45,+1,+1,+1,+1}
{10,67,17,12,45,97,-1,-1,-1,-1}
{43,75,31,93,48,90,+1,+1,+1,+1}
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hejoseph · 发表于 2023-3-29 16:16:25

本帖最后由 hejoseph 于 2023-3-29 16:24 编辑

四面体 $ABCD$ 的体积是 $V$，$AB = a$，$AC = b$，$AD = c$，$CD = p$，$DB = q$，$BC = r$，设
\begin{align*}
P_1 &= (ap)^2(-a^2 + b^2 + c^2 - p^2 + q^2 + r^2) ， \\
P_2 &= (bq)^2(a^2 - b^2 + c^2 + p^2 - q^2 + r^2) ， \\
P_3 &= (cr)^2(a^2 + b^2 - c^2 + p^2 + q^2 - r^2) ， \\
P &= (abr)^2 + (acq)^2 + (bcp)^2 + (pqr)^2 ，
\end{align*}
则
\[
V = \frac{1}{12}\sqrt{P_1 + P_2 + P_3 - P} 。
\]

如果 $P_1 + P_2 + P_3 > P$。设三边长是 $a$、$b$、$r$ 的三角形中边长为$r$所对的内角是 $\alpha_1$，三边长是 $a$、$c$、$q$ 的三角形中边长为 $q$ 所对的内角是 $\alpha_2$，三边长是 $b$、$c$、$p$ 的三角形中边长为 $p$ 所对的内角是 $\alpha_3$，$\theta = (\alpha_1 + \alpha_2 + \alpha_3)/2$，则 $0 < \theta < 270^\circ$，$-90^\circ < \theta - \alpha_1 < 180^\circ$，$-90^\circ < \theta - \alpha_2 < 180^\circ$，$-90^\circ < \theta - \alpha_3 < 180^\circ$，并且
\[
P_1 + P_2 + P_3 - P = 16a^2b^2c^2\sin \theta \sin(\theta - \alpha_1) \sin(\theta - \alpha_2) \sin(\theta - \alpha_3) ，
\]
所以 $\sin \theta \sin(\theta - \alpha_1) \sin(\theta - \alpha_2) \sin(\theta - \alpha_3)$ 必须是正数。

首先 $\sin(\theta - \alpha_1)$、$\sin(\theta - \alpha_2)$、$\sin(\theta - \alpha_3)$ 中不能有两个为负，否则不妨设 $\sin(\theta - \alpha_1) < 0$，$\sin(\theta - \alpha_2) < 0$，则 $\theta - \alpha_1 < 0^\circ$，$\theta - \alpha_2 < 0^\circ$，这样就得到 $\alpha_3 < 0^\circ$，这是不可能的。

另外 $\sin \theta$ 也不能为负，否则 $\sin(\theta - \alpha_1)$、$\sin(\theta - \alpha_2)$、$\sin(\theta - \alpha_3)$ 中有且只有一项必须为负，不妨假设 $\sin(\theta - \alpha_1) < 0$。由 $\sin \theta < 0$ 得 $\alpha_1 + \alpha_2 + \alpha_3 > 360^\circ$；由 $\sin(\theta - \alpha_1) < 0$ 得到 $\alpha_2 + \alpha_3 < \alpha_1 < 180^\circ$，又得 $\alpha_1 + \alpha_2 + \alpha_3 < 360^\circ$，与 $\alpha_1 + \alpha_2 + \alpha_3 > 360^\circ$ 矛盾。

所以 $\sin \theta$、$\sin(\theta - \alpha_1)$、$\sin(\theta - \alpha_2)$、$\sin(\theta - \alpha_3)$ 必须全部为正，这样就得到 $\alpha_1 + \alpha_2 + \alpha_3 < 360^\circ$，$\alpha_1 + \alpha_2 > \alpha_3$，$\alpha_1 + \alpha_3 > \alpha_2$，$\alpha_2 + \alpha_3 > \alpha_1$，由面角能构成三面角的充要条件知点 $A$、$B$、$C$、$D$ 能构成四面体。

nyy · 发表于 2023-3-30 08:21:40

hejoseph 发表于 2023-3-29 16:16
四面体 $ABCD$ 的体积是 $V$，$AB = a$，$AC = b$，$AD = c$，$CD = p$，$DB = q$，$BC = r$，设
\begin{a ...

你的论证我不明白：你说四个正弦都必须正，但是体积表达式大于零的时候，有可能四个面都不构成三角形呀，这时候四个正弦也是正数吗？这个时候估计都没正弦值了，也许是虚数了，或者是大于1的正弦值了。

nyy · 发表于 2023-3-31 10:36:40

hejoseph 发表于 2023-3-29 16:16
四面体 $ABCD$ 的体积是 $V$，$AB = a$，$AC = b$，$AD = c$，$CD = p$，$DB = q$，$BC = r$，设
\begin{a ...

如果能构成三角形，那体积肯定大于零呀！

hejoseph · 发表于 2023-3-31 11:27:51

对于 $AB=6$，$AC=BC=7$，$AD=8$，$BD=CD=4$ 画了个四面体的展开图，其中 $A_1E$ 垂直于 $BC$，$A_2F$ 垂直于 $CD$，$A_3G$ 垂直于 $DB$，三直线 $A_1E$、$A_2F$、$A_3G$ 相交于同一点 $H$，这个点 $H$，这个点 $H$ 就是点 $A$ 在平面 $BCD$ 内的正射影。如果这个四面体存在，那么显然必须 $A_1E>HE$，$A_2F>HF$，$A_3G>HG$，但此时显然与这个要求不符，所以这样的四面体并不存在。

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[提问] abcxyz对应的四面体体积公式大于零，那么abcxyz能构成四面体吗？

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