这个初中几何题如何求解?

王守恩

mathe 发表于 2018-3-5 19:39
对于一般情况，我们假设AA'分角A为u,v两个角，于是有
$tan(v)=tan(A-u)=\frac{tan(A)-tan(u)}{1+tan(A)tan ...

我还是在想：
1，此题先人应该有成熟的解法(先人比我们聪明多了)。
2，可以这样认为(?)：任意一个三角形AEF都恰好对应另一个三角形ABC。

mathematica

1. FullSimplify@Minimize[{Sqrt[b^2+(a-c)^2],(a-4)^2==a^2+d^2&&b^2+(c-4)^2==(b-d)^2+c^2&&b/3+c/4==1},{a,b,c,d}]
   

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我以为直接开放会降低求解速度,没想到也能快速求解出来

\[\left\{\frac{15}{\sqrt{24 \sqrt{6}+59}},\left\{a\to 2 \sqrt{6}-3,b\to 3 \left(\sqrt{6}-2\right),c\to -4 \left(\sqrt{6}-3\right),d\to 2 \left(\sqrt{6}-2\right)\right\}\right\}\]

mathematica

王守恩 发表于 2018-3-10 19:06
我还是在想：
1，此题先人应该有成熟的解法(先人比我们聪明多了)。
2，可以这样认为(?)：任意一个三 ...

利用三次余弦定理,照样可以得到约束,然后求解最小值

1. (*计算折纸的痕迹的最小长度*)
   
2. Clear["Global`*"];(*Clear all variables*)
   
3. FullSimplify@Minimize[
   
4. {g,(*目标函数*)
   
5. (*约束条件*)
   
6. a+b==4&&
   
7. c+d==3&&
   
8. e+f==5&&
   
9. b^2+c^2==a^2&&(*勾股定理或者余弦定理*)
   
10. (a^2+f^2-g^2)/(2*a*f)==4/5&&(*余弦定理*)
    
11. (d^2+e^2-f^2)/(2*d*e)==3/5&&(*余弦定理*)
    
12. a>=0&&
    
13. b>=0&&
    
14. c>=0&&
    
15. d>=0&&
    
16. e>=0&&
    
17. f>=0&&
    
18. g>=0
    
19. },
    
20. {a,b,c,d,e,f,g}](*自变量*)
    

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求解结果

\[\left\{\frac{15}{\sqrt{24 \sqrt{6}+59}},\left\{a\to 7-2 \sqrt{6},b\to 2 \sqrt{6}-3,c\to 2 \left(\sqrt{6}-2\right),d\to 7-2 \sqrt{6},e\to -5 \left(\sqrt{6}-3\right),f\to 5 \left(\sqrt{6}-2\right),g\to \frac{15}{\sqrt{24 \sqrt{6}+59}}\right\}\right\}\]

变量含义见附件图片

1. Clear["Global`*"];(*Clear all variables*)
   
2. FullSimplify@Solve[
   
3. a+b==4&&
   
4. c+d==3&&
   
5. e+f==5&&
   
6. b^2+c^2==a^2&&(*勾股定理或者余弦定理*)
   
7. (a^2+f^2-g^2)/(2*a*f)==4/5&&(*余弦定理*)
   
8. (d^2+e^2-f^2)/(2*d*e)==3/5&&(*余弦定理*)
   
9. a>=0&&
   
10. b>=0&&
    
11. c>=0&&
    
12. d>=0&&
    
13. e>=0&&
    
14. f>=0&&
    
15. g>=0
    
16. ,
    
17. {a,b,d,e,f,g}(*自变量*)
    
18. ]
    

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求解出各个长度以c为变量
\[\left\{\left\{a\to \text{ConditionalExpression}\left[\frac{c^2}{8}+2,0<c<3\right],b\to \text{ConditionalExpression}\left[2-\frac{c^2}{8},0<c<3\right],d\to \text{ConditionalExpression}[3-c,0<c<3],e\to \text{ConditionalExpression}\left[-\frac{5 (c-8) (c+2)}{6 c+32},0<c<3\right],f\to \text{ConditionalExpression}\left[\frac{5 \left(c^2+16\right)}{6 c+32},0<c<3\right],g\to \text{ConditionalExpression}\left[\frac{3 \left(c^2+16\right)^{3/2}}{8 (3 c+16)},0<c<3\right]\right\}\right\}\]

mathematica

mathematica 发表于 2018-3-22 10:03
利用三次余弦定理,照样可以得到约束,然后求解最小值

1. Clear["Global`*"];(*Clear all variables*)
   
2. FullSimplify@Solve[
   
3. a+b==4&&
   
4. c+d==3&&
   
5. e+f==5&&
   
6. b^2+c^2==a^2&&(*勾股定理或者余弦定理*)
   
7. (a^2+f^2-g^2)/(2*a*f)==4/5&&(*余弦定理*)
   
8. (d^2+e^2-f^2)/(2*d*e)==3/5&&(*余弦定理*)
   
9. a>=0&&
   
10. b>=0&&
    
11. c>=0&&
    
12. d>=0&&
    
13. e>=0&&
    
14. f>=0&&
    
15. g>=0
    
16. ,
    
17. {a,b,c,d,e,f}(*自变量*)
    
18. ]
    

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查看共同定义域的最小值就可以了
\[\left\{\left\{a\to \text{ConditionalExpression}\left[\frac{1}{8} \left(\left(\text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,1\right]-6\right) \text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,1\right]+25\right),\frac{3}{2}<g<\frac{15}{8}\lor \frac{15}{\sqrt{24 \sqrt{6}+59}}<g<\frac{3}{2}\right],b\to \text{ConditionalExpression}\left[-\frac{1}{8} \left(\text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,1\right]-7\right) \left(\text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,1\right]+1\right),\frac{3}{2}<g<\frac{15}{8}\lor \frac{15}{\sqrt{24 \sqrt{6}+59}}<g<\frac{3}{2}\right],c\to \text{ConditionalExpression}\left[3-\text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,1\right],\frac{3}{2}<g<\frac{15}{8}\lor \frac{15}{\sqrt{24 \sqrt{6}+59}}<g<\frac{3}{2}\right],d\to \text{ConditionalExpression}\left[\text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,1\right],\frac{3}{2}<g<\frac{15}{8}\lor \frac{15}{\sqrt{24 \sqrt{6}+59}}<g<\frac{3}{2}\right],e\to \text{ConditionalExpression}\left[\frac{5 \left(\text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,1\right]^2-25\right)}{6 \text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,1\right]-50},\frac{3}{2}<g<\frac{15}{8}\lor \frac{15}{\sqrt{24 \sqrt{6}+59}}<g<\frac{3}{2}\right],f\to \text{ConditionalExpression}\left[\frac{5}{18} \left(-3 \text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,1\right]+\frac{400}{25-3 \text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,1\right]}-7\right),\frac{3}{2}<g<\frac{15}{8}\lor \frac{15}{\sqrt{24 \sqrt{6}+59}}<g<\frac{3}{2}\right]\right\},\left\{a\to \text{ConditionalExpression}\left[\frac{1}{8} \left(\left(\text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,2\right]-6\right) \text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,2\right]+25\right),\frac{15}{\sqrt{24 \sqrt{6}+59}}<g<\frac{3}{2}\right],b\to \text{ConditionalExpression}\left[-\frac{1}{8} \left(\text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,2\right]-7\right) \left(\text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,2\right]+1\right),\frac{15}{\sqrt{24 \sqrt{6}+59}}<g<\frac{3}{2}\right],c\to \text{ConditionalExpression}\left[3-\text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,2\right],\frac{15}{\sqrt{24 \sqrt{6}+59}}<g<\frac{3}{2}\right],d\to \text{ConditionalExpression}\left[\text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,2\right],\frac{15}{\sqrt{24 \sqrt{6}+59}}<g<\frac{3}{2}\right],e\to \text{ConditionalExpression}\left[\frac{5 \left(\text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,2\right]^2-25\right)}{6 \text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,2\right]-50},\frac{15}{\sqrt{24 \sqrt{6}+59}}<g<\frac{3}{2}\right],f\to \text{ConditionalExpression}\left[\frac{5}{18} \left(-3 \text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,2\right]+\frac{400}{25-3 \text{Root}\left[9 \text{$\#$1}^6-162 \text{$\#$1}^5+1647 \text{$\#$1}^4-10044 \text{$\#$1}^3+\left(41175-576 g^2\right) \text{$\#$1}^2+\left(9600 g^2-101250\right) \text{$\#$1}-40000 g^2+140625\&,2\right]}-7\right),\frac{15}{\sqrt{24 \sqrt{6}+59}}<g<\frac{3}{2}\right]\right\}\right\}\]


可知道结果是
\[\frac{15}{\sqrt{24 \sqrt{6}+59}}\]