这道初中还是高中几何题有简单解法吗？ - 第3页 - 小题大作 - 数学研发论坛

dlpg070 发表于 2020-5-25 16:45:32

本帖最后由 dlpg070 于 2020-5-25 16:54 编辑

dlpg070 发表于 2020-5-23 16:43
你的公式正确,给出验证代码和计算结果
输出:
最小值 FE=7.07107 CE=5. CF=5.

我的回复真么没有显示,重发

EF最小值计算公式证明参考图:
矩形最小值_8_11.png
公式:
在 CE=CF=a+b-Sqrt(2*a*b)时
EF最小值=Sqrt*(a+b-Sqrt(2*a*b))

证明中利用三角公式演算求极值,与坐标无关
考虑到对称性和边界条件,只讨论 0<a<=1/2*b
显然:
角EAD+角FAB=45度

Tan[角EAD]= (a-CE)/b
Tan[角FAB]= (b-CF)/a
所以
ArcTan[(a-CE)/b] + ArcTan[(b-CF)/a] = 45 *Pi/180

下面是计算机辅助证明的代码:
(* ArcTan[(a-CE)/b]+ArcTan[(b-CF)/a]]= Pi/4
求解 EF最小值的符号表达式
*)
Clear["Global`*"]
aa=(a-CE/b);
bb=(b-CF)/a;
tt=TrigExpand[ Tan[(Pi/4- ArcTan[(a-CE)/b])]]//FullSimplify;
CF=b-a* tt;(*CF的符号表达式 CF *)
Print["CF= ",CF]
EF2=CE^2+CF^2// FullSimplify;
sol=Solve==0 ,{CE}];
CEmin=CE/.sol[];(*第一个解是最小值 *)
CE=CEmin;
CFmin=CF //FullSimplify;
CF=CFmin;
EFmin=Sqrt;
Print["最小值 \nFE=",EFmin,", \nCE=",CEmin,", \nCF=",CFmin]
Print["=== end ==="]

CF= b-(a (-a+b+CE))/(a+b-CE)
最小值
FE=Sqrt Sqrt[(a-Sqrt Sqrt Sqrt+b)^2],
CE=a-Sqrt Sqrt Sqrt+b,
CF=a-Sqrt Sqrt Sqrt+b
=== end ===

dlpg070 发表于 2020-5-26 16:06:05

本帖最后由 dlpg070 于 2020-5-26 16:08 编辑

王守恩发表于 2020-5-24 07:55
继续猜想！接12楼。

记：\(\D\tan^{-1}\bigg(\frac{AB-CE(45-k)^\circ}{AD}\bigg)+\tan^{-1}\bigg(\fr ...

角EAF=60度EF最小值符号表达式解
应王守恩要求发布,供参考,不如数值解实用

CF= b-a (-Sqrt+(4 b)/(a+Sqrt b-CE))
最小值
FE=\((1/8 (6 a+6 Sqrt b)-1/2 \(a^2/4+1/2 Sqrt a b+(3 b^2)/4+(4 2^(1/3) Sqrt a^3 b)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)-(48 2^(1/3) a^2 b^2)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)+(4 2^(1/3) Sqrt a b^3)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)+(1/(3 2^(1/3)))((864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)))-1/2 \(-((25 a^2)/4)-25/2 Sqrt a b-(75 b^2)/4+3/16 (-6 a-6 Sqrt b)^2-(4 2^(1/3) Sqrt a^3 b)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)+(48 2^(1/3) a^2 b^2)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)-(4 2^(1/3) Sqrt a b^3)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)-(1/(3 2^(1/3)))((864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3))-(-4 (-2 a^3-14 Sqrt a^2 b-26 a b^2-6 Sqrt b^3)+1/2 (-6 a-6 Sqrt b) (-(1/4) (-6 a-6 Sqrt b)^2+2 (6 a^2+12 Sqrt a b+18 b^2)))/(4 \(a^2/4+1/2 Sqrt a b+(3 b^2)/4+(4 2^(1/3) Sqrt a^3 b)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)-(48 2^(1/3) a^2 b^2)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)+(4 2^(1/3) Sqrt a b^3)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)+(1/(3 2^(1/3)))((864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3))))))^2+(b-a (-Sqrt+(4 b)/(a+Sqrt b+1/8 (-6 a-6 Sqrt b)+1/2 \(a^2/4+1/2 Sqrt a b+(3 b^2)/4+(4 2^(1/3) Sqrt a^3 b)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)-(48 2^(1/3) a^2 b^2)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)+(4 2^(1/3) Sqrt a b^3)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)+(1/(3 2^(1/3)))((864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)))+1/2 \(-((25 a^2)/4)-25/2 Sqrt a b-(75 b^2)/4+3/16 (-6 a-6 Sqrt b)^2-(4 2^(1/3) Sqrt a^3 b)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)+(48 2^(1/3) a^2 b^2)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)-(4 2^(1/3) Sqrt a b^3)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)-(1/(3 2^(1/3)))((864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3))-(-4 (-2 a^3-14 Sqrt a^2 b-26 a b^2-6 Sqrt b^3)+1/2 (-6 a-6 Sqrt b) (-(1/4) (-6 a-6 Sqrt b)^2+2 (6 a^2+12 Sqrt a b+18 b^2)))/(4 \(a^2/4+1/2 Sqrt a b+(3 b^2)/4+(4 2^(1/3) Sqrt a^3 b)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)-(48 2^(1/3) a^2 b^2)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)+(4 2^(1/3) Sqrt a b^3)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)+(1/(3 2^(1/3)))((864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3))))))))^2),
CE=1/8 (6 a+6 Sqrt b)-1/2 \(a^2/4+1/2 Sqrt a b+(3 b^2)/4+(4 2^(1/3) Sqrt a^3 b)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)-(48 2^(1/3) a^2 b^2)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)+(4 2^(1/3) Sqrt a b^3)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)+(1/(3 2^(1/3)))((864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)))-1/2 \(-((25 a^2)/4)-25/2 Sqrt a b-(75 b^2)/4+3/16 (-6 a-6 Sqrt b)^2-(4 2^(1/3) Sqrt a^3 b)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)+(48 2^(1/3) a^2 b^2)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)-(4 2^(1/3) Sqrt a b^3)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)-(1/(3 2^(1/3)))((864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3))-(-4 (-2 a^3-14 Sqrt a^2 b-26 a b^2-6 Sqrt b^3)+1/2 (-6 a-6 Sqrt b) (-(1/4) (-6 a-6 Sqrt b)^2+2 (6 a^2+12 Sqrt a b+18 b^2)))/(4 \(a^2/4+1/2 Sqrt a b+(3 b^2)/4+(4 2^(1/3) Sqrt a^3 b)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)-(48 2^(1/3) a^2 b^2)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)+(4 2^(1/3) Sqrt a b^3)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)+(1/(3 2^(1/3)))((864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3))))),
CF=b-a (-Sqrt+(4 b)/(a+Sqrt b+1/8 (-6 a-6 Sqrt b)+1/2 \(a^2/4+1/2 Sqrt a b+(3 b^2)/4+(4 2^(1/3) Sqrt a^3 b)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)-(48 2^(1/3) a^2 b^2)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)+(4 2^(1/3) Sqrt a b^3)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)+(1/(3 2^(1/3)))((864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)))+1/2 \(-((25 a^2)/4)-25/2 Sqrt a b-(75 b^2)/4+3/16 (-6 a-6 Sqrt b)^2-(4 2^(1/3) Sqrt a^3 b)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)+(48 2^(1/3) a^2 b^2)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)-(4 2^(1/3) Sqrt a b^3)/(864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3)-(1/(3 2^(1/3)))((864 a^4 b^2-864 a^2 b^4+Sqrt[-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2])^(1/3))-(-4 (-2 a^3-14 Sqrt a^2 b-26 a b^2-6 Sqrt b^3)+1/2 (-6 a-6 Sqrt b) (-(1/4) (-6 a-6 Sqrt b)^2+2 (6 a^2+12 Sqrt a b+18 b^2)))/(4 \(a^2/4+1/2 Sqrt a b+(3 b^2)/4+(4 2^(1/3) Sqrt a^3 b)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)-(48 2^(1/3) a^2 b^2)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)+(4 2^(1/3) Sqrt a b^3)/(864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)+(1/(3 2^(1/3)))((864 a^4 b^2-864 a^2 b^4+\(-6912 (Sqrt a^3 b-12 a^2 b^2+Sqrt a b^3)^3+(864 a^4 b^2-864 a^2 b^4)^2))^(1/3)))))))
=== end ===

王守恩 发表于 2020-5-26 20:14:17

本帖最后由王守恩于 2020-5-26 20:15 编辑

dlpg070 发表于 2020-5-26 16:06
角EAF=60度EF最小值符号表达式解
应王守恩要求发布,供参考,不如数值解实用

角EAF=60度EF最小值符号表达式解
换个公式(不用三角函数)，符号表达式解还是会这样吓人吗？
已知：\(\D(AB)^2+(BF)^2+(AD)^2+(DE)^2-\sqrt{\bigg((AB)^2+(BF)^2\bigg)\bigg((AD)^2+(DE)^2\bigg)}=(EC)^2+(CF)^2\)
求：\(\D\sqrt{(EC)^2+(CF)^2}\) 的最小值

mathematica 发表于 2020-5-27 08:18:52

王守恩发表于 2020-5-26 20:14
角EAF=60度EF最小值符号表达式解
换个公式(不用三角函数)，符号表达式解还是会这样吓人吗？
已知 ...

弄个具体的吧，符号太难了！

dlpg070 发表于 2020-5-27 08:59:46

本帖最后由 dlpg070 于 2020-5-27 09:17 编辑

王守恩发表于 2020-5-26 20:14
角EAF=60度EF最小值符号表达式解
换个公式(不用三角函数)，符号表达式解还是会这样吓人吗？
已知 ...

利用你的等式(实际是余弦定理)也许会简单点,但仍很复杂
关键是最小值求不出来,你的那些关系式怎知是否正确?
初步分析,问题相当复杂,一般情况下你12#猜想不成立
不如用具体数值例子来验证或反证
以原题的a=2 b=4 为例

角EAF=60度最小值 EF 及 CE CF记为 EF(60),CE(60,CF(60)
角EAF=30度最小值 EF 及 CE CF记为 EF(30),CE(30,CF(30)
比较 CE(30)/CF(30) == CE(60)/CF(60)
你我和感兴趣的一起算一算,期待结果 !
我的结果将分 60 30分别贴出

王守恩 发表于 2020-5-27 09:53:07

本帖最后由王守恩于 2020-5-27 10:25 编辑

dlpg070 发表于 2020-5-27 08:59
利用你的等式(实际是余弦定理)也许会简单点,但仍很复杂
关键是最小值求不出来,你的那些关系式怎知是否 ...

利用 Minimize 可以少立一个方程，可惜 Minimize 与 NMinimize 差别蛮大的，
数值解规律不好找，我就是想搞个通项出来(跟45°一样)

mathematica 发表于 2020-7-22 15:12:18

有一个模型叫做探照灯模型，用相似三角形的办法，等角等高找等腰，然后取最小值吧

许言发表于 2020-9-10 09:05:08

mathematica 发表于 2020-5-21 11:55
有没有办法证明CEF必然是等腰直角三角形
对于直角三角形CEF，有EF2=CE2+CF2>=2CE*CF,等号在CE=CF取得，易证可取得

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