中秋画月饼
中秋画月饼中秋吃月饼确实惬意,奈何血糖偏高,唯有望"月"兴叹
突发奇想,画几个月饼,打发时光
问题:三角形的月饼盒放3个大小不同的圆月饼,月饼的面积sc与月饼盒的面积st之比 k=sc/st,求k最大值?
凝思良久,不得要领,你愿来茶馆聊聊? 正三角形内三个相等的圆？ 本帖最后由 aimisiyou 于 2020-10-3 18:29 编辑
markfang2050 发表于 2020-10-3 17:16
正三角形内三个相等的圆？
左边三个圆面积和大于右边三个圆的面积和。猜测对于任意三角形，三个圆分别为三角形内接圆及该圆两个相邻较大区域内的内接圆。 堆砌问题是颠峰级别的几何题 本帖最后由 dlpg070 于 2020-10-4 14:30 编辑
aimisiyou 发表于 2020-10-3 18:21
左边三个圆面积和大于右边三个圆的面积和。猜测对于任意三角形，三个圆分别为三角形内接圆及该圆两个相 ...
按你的猜想,大园尽可能大,取为内切圆的方案,做了11例子,边解题边画图,题解完了,图也画完了
有趣的是:k最大值出现在角A=78度附近的等腰三角形,待更多的测试,k能大于0.762232 ?
大园最大方案解题结果:
例1 a=10 b=10 c=10 等边三角形 k=0.738955
例2 a=10 b=6c=6钝角三角形 k=0.692925
例3 a=10 b=9c=6锐角三角形 k=0.736986
例4 a=5b=4c=3直角三角形 k=0.741296
例5 a=Sqrt b=5 c=1 直角三角形 k=0.43955接近最小?
例6 a=10 b=5 Sqrt c=5 Sqrt 等腰直角三角形 k=0.753893 A=90度
例7 a=9 b=5 Sqrt c=5 Sqrt7等腰锐角三角形 k=0.762194 aA= 79.04721580度
例11 a=8.9 b=5 Sqrt c=5 Sqrt 11等腰钝角三角形 k=0.762232 aA= 78.0007度 接近最大?
例8 a=8 b=5 Sqrt c=5 Sqrt8等腰锐角三角形 k=0.756459
例9 a=7 b=5 Sqrt c=5 Sqrt9等腰锐角三角形 k=0.737119 A接近60度
例10 a=11 b=5 Sqrt c=5 Sqrt10等腰钝角三角形 k=0.728891
图形文件名:画月饼_例4_a5_b4_c3_k0.741296-20201003.png
在上图中，设BD=p,DC=q, AE=r, OD=1, 左边小圆半径为u,右边小圆半径为v，记s=p+q+r
计算可得， $p=\frac{\sqrt(4u)}{1-u}, q=\frac{\sqrt{4v}}{1-v}$
根据海伦公式,$1=\sqrt{\frac{pqr}s}$, 即$s=p+q+r=pqr$,所以三角形ABC的面积为$\sqrt{pqrs}=pqr$
而且$r=\frac{p+q}{pq-1}$, 得到三角形ABC的面积为$\Delta=\frac{pq(p+q)}{pq-1}$, 可以看出必须有$pq\gt 1$.
我们设$u=\tan^2(\frac{\alpha}2), v=\tan^2(\frac{\beta}2)$,于是$0\lt \alpha \lt \frac{\pi}2, 0\lt \beta\lt\frac{\pi}2$
于是得到$p=\tan(\alpha), q=\tan(\beta), \Delta=-\tan(\alpha)\tan(\beta)\tan(\alpha+\beta))$ 其中条件$pq\gt 1$转化为$\alpha+\beta \gt \frac{\pi}2$
而三个圆的面积和为$\pi(1+u+v)=\pi(1+\tan^4(\frac{\alpha}2)+\tan^4(\frac{\beta}2)$
于是问题就转化为给定
$0\lt \alpha\lt\frac{\pi}2, 0\lt \beta\lt\frac{\pi}2, \alpha+\beta \gt \frac{\pi}2$
求$-\frac{\pi (1+\tan^4(\frac{\alpha}2)+\tan^4(\frac{\beta}2))}{\tan(\alpha)\tan(\beta)\tan(\alpha+\beta)}$的最大值 本帖最后由 dlpg070 于 2020-10-4 18:38 编辑
再给出一些实例数据,供理论分析参考
从边长10的等边三角形的A向下压扁,a不变,b c逐渐变小,
k的变化:
注意最大值 最小值
n a b c A 度 k
0 10 10.00 10.00 60.000000 0.73895529653986653172
1 10 9.950 9.950 60.332739 0.73984646402223665662
2 10 9.900 9.900 60.669410 0.74072832625895633803
3 10 9.850 9.850 61.010088 0.74160052418456170601
4 10 9.800 9.800 61.354849 0.74246268644904800242
5 10 9.750 9.750 61.703772 0.74331442890621542114
6 10 9.700 9.700 62.056938 0.74415535407522913650
7 10 9.650 9.650 62.414430 0.74498505057367113103
8 10 9.600 9.600 62.776333 0.74580309252022981217
9 10 9.550 9.550 63.142735 0.74660903890503015905
10 10 9.500 9.500 63.513728 0.74740243292545109798
11 10 9.450 9.450 63.889403 0.74818280128510667560
12 10 9.400 9.400 64.269856 0.74894965345348193927
13 10 9.350 9.350 64.655186 0.74970248088351163905
14 10 9.300 9.300 65.045495 0.75044075618416814205
15 10 9.250 9.250 65.440887 0.75116393224488229633
16 10 9.200 9.200 65.841469 0.75187144130835515825
17 10 9.150 9.150 66.247353 0.75256269398802699838
18 10 9.100 9.100 66.658653 0.75323707822615001606
19 10 9.050 9.050 67.075487 0.75389395818805956679
20 10 9.000 9.000 67.497977 0.75453267308785189580
21 10 8.950 8.950 67.926249 0.75515253594025038469
22 10 8.900 8.900 68.360431 0.75575283223297266010
23 10 8.850 8.850 68.800659 0.75633281851339251139
24 10 8.800 8.800 69.247071 0.75689172088271768333
25 10 8.750 8.750 69.699809 0.75742873339027076219
26 10 8.700 8.700 70.159022 0.75794301631975820512
27 10 8.650 8.650 70.624862 0.75843369435863372472
28 10 8.600 8.600 71.097488 0.75889985464079724333
29 10 8.550 8.550 71.577063 0.75934054465190866723
30 10 8.500 8.500 72.063758 0.75975476998552447243
31 10 8.450 8.450 72.557748 0.76014149193707045834
32 10 8.400 8.400 73.059214 0.76049962492132989379
33 10 8.350 8.350 73.568347 0.76082803369763417086
34 10 8.300 8.300 74.085342 0.76112553038527176785
35 10 8.250 8.250 74.610402 0.76139087124975637356
36 10 8.200 8.200 75.143739 0.76162275323848827221
37 10 8.150 8.150 75.685571 0.76181981024197198809
38 10 8.100 8.100 76.236129 0.76198060905408006301
39 10 8.050 8.050 76.795647 0.76210364500183397756
40 10 8.000 8.000 77.364375 0.76218733721175780314
41 10 7.950 7.950 77.942568 0.76223002347598896254 最大值
42 10 7.900 7.900 78.530496 0.76222995467693430369
43 10 7.850 7.850 79.128437 0.76218528872425756349
44 10 7.800 7.800 79.736683 0.76209408395228115369
45 10 7.750 7.750 80.355539 0.76195429191936917958
46 10 7.700 7.700 80.985323 0.76176374954339773205
47 10 7.650 7.650 81.626368 0.76152017049885664502
48 10 7.600 7.600 82.279021 0.76122113579127885517
49 10 7.550 7.550 82.943648 0.76086408341333884395
50 10 7.500 7.500 83.620630 0.76044629697383734720
51 10 7.450 7.450 84.310369 0.75996489317558067291
52 10 7.400 7.400 85.013285 0.75941680800049131734
53 10 7.350 7.350 85.729820 0.75879878143969623521
54 10 7.300 7.300 86.460442 0.75810734058227872799
55 10 7.250 7.250 87.205638 0.75733878084818015239
56 10 7.200 7.200 87.965926 0.75648914511758292692
57 10 7.150 7.150 88.741852 0.75555420046999883913
58 10 7.100 7.100 89.533991 0.75452941220000066853
59 10 7.050 7.050 90.342952 0.75340991472155830088
60 10 7.000 7.000 91.169383 0.75219047890739844783
61 10 6.950 6.950 92.013966 0.75086547533136573853
62 10 6.900 6.900 92.877431 0.74942883278750141342
63 10 6.850 6.850 93.760550 0.74787399134579984051
64 10 6.800 6.800 94.664149 0.74619384906670541840
65 10 6.750 6.750 95.589107 0.74438070132846179301
66 10 6.700 6.700 96.536366 0.74242617151585702280
67 10 6.650 6.650 97.506933 0.74032113156598051306
68 10 6.600 6.600 98.501891 0.73805561055369057376
69 10 6.550 6.550 99.522404 0.73561868911009616846
70 10 6.500 6.500 100.56973 0.73299837697979365039
71 10 6.450 6.450 101.64521 0.73018147040811387659
72 10 6.400 6.400 102.75033 0.72715338526978315095
73 10 6.350 6.350 103.88669 0.72389796085332581211
74 10 6.300 6.300 105.05601 0.72039722793061939884
75 10 6.250 6.250 106.26020 0.71663113307111068399
76 10 6.200 6.200 107.50136 0.71257720897021907746
77 10 6.150 6.150 108.78177 0.70821017766132775963
78 10 6.100 6.100 110.10397 0.70350146959971941620
79 10 6.050 6.050 111.47079 0.69841863635385937081
80 10 6.000 6.000 112.88538 0.69292462744167046428
81 10 5.950 5.950 114.35126 0.68697689185405211373
82 10 5.900 5.900 115.87243 0.68052625072320953470
83 10 5.850 5.850 117.45340 0.67351546742578177262
84 10 5.800 5.800 119.09937 0.66587741201878897318
85 10 5.750 5.750 120.81631 0.65753267322498961457
86 10 5.700 5.700 122.61117 0.64838640483655650387
87 10 5.650 5.650 124.49215 0.63832409013909689317
88 10 5.600 5.600 126.46900 0.62720574273363746661
89 10 5.550 5.550 128.55348 0.61485778934498072542
90 10 5.500 5.500 130.76005 0.60106141326219990770
91 10 5.450 5.450 133.10677 0.58553530349420852316
92 10 5.400 5.400 135.61679 0.56790919117421328771
93 10 5.350 5.350 138.32061 0.54768144270422178941
94 10 5.300 5.300 141.25992 0.52414731893238520485
95 10 5.250 5.250 144.49442 0.49626889625412479980
96 10 5.200 5.200 148.11526 0.46241643524759646403
97 10 5.150 5.150 152.27514 0.41978302109499773548
98 10 5.100 5.100 157.27025 0.36276825559872228290
99 10 5.050 5.050 163.86140 0.27656865045132868415 最小值
可以在我上面推导的表达式中取$\alpha=\beta\gt \frac{\pi}4$,得到函数在$\alpha=1.1267645149718208888226682817341231170$时可以取到最大值
$0.76223544633708454009202606640962263987$
对应p=q=2.1020975301232264318317503841814911767,r=1.2297232397226004394477823877671332852
三角形三边长,p+r=q+r=3.3318207698458268712795327719486244618,p+q=4.2041950602464528636635007683629823534
内切圆半径1，两小圆半径u=0.39900859647976833900653276659241113097 本帖最后由 dlpg070 于 2020-10-5 09:05 编辑
mathe 发表于 2020-10-5 08:16
可以在我上面推导的表达式中取$\alpha=\beta\gt \frac{\pi}4$,得到函数在$\alpha=1.1267645149718208888226 ...
我的初步数值计算结果和你的理论结果相同
当前最大值点 a : b : c = 1:0.792500 : 0.792500
我还在找最大值点的 a : b : c 是多少
mathe理论结果:alpha=1.1267645149718208888226682817341231170 k=0.76223544633708454009202606640962263987
我的数值结果 : a=10 b=7.92500 c=7.92500 A=78.235298度 k=0.76223544633636543687 *最大值 加密计算
本帖最后由 dlpg070 于 2020-10-5 14:41 编辑
mathe 发表于 2020-10-5 08:16
可以在我上面推导的表达式中取$\alpha=\beta\gt \frac{\pi}4$,得到函数在$\alpha=1.1267645149718208888226 ...
利用你的例子三角形验算,并且画图
显示
mathe的例子的三角形和我的例11 是相似三角形,结果除尺寸大小不同外相同,计算结果的差异纯属误差
mathe k=0.76223544633708454009202606640962263987
我的最新结果 k=0.76223544633707707293908762187
小数点后13位相同
例12 a=4.2041950602464528636635007683629823534 b=3.3318207698458268712795327719486244618 c=3.331820769845826871279532771948624461812mathe的例子三角形
详细结果:
aA= 78.23540485度
aB= 50.88229757度
aC= 50.88229757度
A= {2.102097530123226431831750384181491177,2.584998184956010675911694180571155815}
B= {0,0}
C= {4.2041950602464528636635007683629823534,0}
D= {2.102097530123226431831750384181491177,0}
E= {2.87794904301106587317004310671404825,1.63091554898389587023823877802206685}
F= {1.32624601723538699049345766164893410,1.63091554898389587023823877802206685}
外心O1= {2.1020975301232264318317503841814911767,0.43779558594046312395993297282784981}
重心O= {2.102097530123226431831750384181491177,0.861666061652003558637231393523718605}
大园圆心= {2.102097530123226431831750384181491177,1.000000000000000000000000000000000000} 半径=1.
中园圆心= {3.36544,0.399009} 半径=0.399009(为演示,精度不高)
小园圆心= {0.838755,0.399009} 半径=0.399009(为演示,精度不高)
三角形面积 st=5.43391829996905330311128315613011564 园面积和=4.14193 k=0.762235
图形文件名:画月饼_例12_a4.20420_b3.33182_c3.33182_k0.762235-20201003.png