周长,面积相等的不同梯形

northwolves · 发表于 6 天前

{52,{{{6,9,16,21},36 Sqrt[11]},{{4,10,18,20},36 Sqrt[11]},{{5,10,18,19},36 Sqrt[11]}}}
{58,{{{8,12,16,22},45 Sqrt[15]},{{4,15,19,20},45 Sqrt[15]},{{11,12,16,19},45 Sqrt[15]}}}
{60,{{{10,14,16,20},120 Sqrt[3]},{{12,13,15,20},120 Sqrt[3]},{{12,14,16,18},120 Sqrt[3]}}}

{108,{{{14,22,30,42},240 Sqrt[6]},{{18,20,28,42},240 Sqrt[6]},{{9,25,35,39},240 Sqrt[6]},{{3,33,35,37},240 Sqrt[6]}}},
{120,{{{15,29,31,45},120 Sqrt[42]},{{18,26,34,42},120 Sqrt[42]},{{20,26,34,40},120 Sqrt[42]},{{28,29,31,32},120 Sqrt[42]}}}

{184,{{{24,29,56,75},396 Sqrt[13]},{{14,33,63,74},396 Sqrt[13]},{{27,29,56,72},396 Sqrt[13]},{{14,34,66,70},396 Sqrt[13]},{{20,33,63,68},396 Sqrt[13]}}}
{186,{{{23,42,52,69},336 Sqrt[30]},{{13,52,56,65},336 Sqrt[30]},{{21,44,58,63},336 Sqrt[30]},{{25,44,58,59},336 Sqrt[30]},{{29,43,56,58},336 Sqrt[30]}}}}

northwolves · 发表于 6 天前

{236,{{{13,57,75,91},720 Sqrt[14]},{{10,60,76,90},720 Sqrt[14]},{{18,52,76,90},720 Sqrt[14]},{{31,45,71,89},720 Sqrt[14]},{{34,45,71,86},720 Sqrt[14]},{{38,50,72,76},720 Sqrt[14]}}}
{264,{{{42,50,60,112},1320 Sqrt[6]},{{22,62,70,110},1320 Sqrt[6]},{{6,70,84,104},1320 Sqrt[6]},{{32,49,83,100},1320 Sqrt[6]},{{33,49,83,99},1320 Sqrt[6]},{{60,62,70,72},1320 Sqrt[6]}}}

northwolves · 发表于 6 天前

{340,{{{42,74,86,138},2160 Sqrt[7]},{{50,64,96,130},2160 Sqrt[7]},{{35,72,108,125},2160 Sqrt[7]},{{58,64,96,122},2160 Sqrt[7]},{{8,99,113,120},2160 Sqrt[7]},{{44,72,108,116},2160 Sqrt[7]},{{5,109,111,115},2160 Sqrt[7]}}}
{362,{{{25,81,101,155},540 Sqrt[110]},{{27,73,109,153},540 Sqrt[110]},{{37,63,114,148},540 Sqrt[110]},{{38,63,119,142},540 Sqrt[110]},{{41,63,119,139},540 Sqrt[110]},{{64,73,109,116},540 Sqrt[110]},{{76,81,101,104},540 Sqrt[110]}}}

northwolves · 发表于 6 天前

8个的：

{500,{{{85,108,120,187},360 Sqrt[1463]},{{60,124,136,180},360 Sqrt[1463]},{{90,103,127,180},360 Sqrt[1463]},{{96,102,128,174},360 Sqrt[1463]},{{68,116,144,172},360 Sqrt[1463]},{{85,108,137,170},360 Sqrt[1463]},{{96,109,135,160},360 Sqrt[1463]},{{85,116,144,155},360 Sqrt[1463]}}},
{636,{{{62,151,175,248},8400 Sqrt[6]},{{93,125,175,243},8400 Sqrt[6]},{{110,118,168,240},8400 Sqrt[6]},{{66,140,196,234},8400 Sqrt[6]},{{30,174,212,220},8400 Sqrt[6]},{{21,197,199,219},8400 Sqrt[6]},{{118,125,175,218},8400 Sqrt[6]},{{94,140,196,206},8400 Sqrt[6]}}},{760,{{{55,192,211,302},10710 Sqrt[7]},{{40,204,220,296},10710 Sqrt[7]},{{125,136,204,295},10710 Sqrt[7]},{{70,163,240,287},10710 Sqrt[7]},{{65,168,252,275},10710 Sqrt[7]},{{140,142,210,268},10710 Sqrt[7]},{{86,168,252,254},10710 Sqrt[7]},{{107,163,240,250},10710 Sqrt[7]}}},
{768,{{{104,172,180,312},2880 Sqrt[105]},{{84,165,219,300},2880 Sqrt[105]},{{104,152,226,286},2880 Sqrt[105]},{{78,166,242,282},2880 Sqrt[105]},{{24,220,260,264},2880 Sqrt[105]},{{132,152,220,264},2880 Sqrt[105]},{{104,166,242,256},2880 Sqrt[105]},{{144,165,219,240},2880 Sqrt[105]}}},{780,{{{120,174,186,300},5040 Sqrt[42]},{{75,203,217,285},5040 Sqrt[42]},{{138,156,204,282},5040 Sqrt[42]},{{66,205,239,270},5040 Sqrt[42]},{{150,156,204,270},5040 Sqrt[42]},{{96,182,238,264},5040 Sqrt[42]},{{102,183,240,255},5040 Sqrt[42]},{{110,182,238,250},5040 Sqrt[42]}}},
{800,{{{136,147,173,344},5040 Sqrt[33]},{{80,177,203,340},5040 Sqrt[33]},{{96,142,238,324},5040 Sqrt[33]},{{37,181,259,323},5040 Sqrt[33]},{{85,147,245,323},5040 Sqrt[33]},{{12,196,289,303},5040 Sqrt[33]},{{130,142,238,290},5040 Sqrt[33]},{{112,153,255,280},5040 Sqrt[33]}}},
{800,{{{57,181,239,323},2520 Sqrt[143]},{{70,168,240,322},2520 Sqrt[143]},{{64,171,245,320},2520 Sqrt[143]},{{23,212,252,313},2520 Sqrt[143]},{{120,140,228,312},2520 Sqrt[143]},{{103,147,241,309},2520 Sqrt[143]},{{40,184,280,296},2520 Sqrt[143]},{{36,189,287,288},2520 Sqrt[143]}}}

northwolves · 发表于 6 天前

9个的：

{700,{{{130,132,148,290},13860 Sqrt[3]},{{98,144,171,287},13860 Sqrt[3]},{{107,129,186,278},13860 Sqrt[3]},{{42,176,209,273},13860 Sqrt[3]},{{95,134,206,265},13860 Sqrt[3]},{{70,147,223,260},13860 Sqrt[3]},{{20,198,222,260},13860 Sqrt[3]},{{108,134,206,252},13860 Sqrt[3]},{{91,147,223,239},13860 Sqrt[3]}}},
{790,{{{110,160,201,319},11700 Sqrt[7]},{{75,185,215,315},11700 Sqrt[7]},{{135,143,197,315},11700 Sqrt[7]},{{120,149,225,296},11700 Sqrt[7]},{{95,160,240,295},11700 Sqrt[7]},{{35,207,258,290},11700 Sqrt[7]},{{122,156,234,278},11700 Sqrt[7]},{{115,160,240,275},11700 Sqrt[7]},{{170,185,215,220},11700 Sqrt[7]}}}
{812,{{{120,178,186,328},10080 Sqrt[11]},{{98,180,212,322},10080 Sqrt[11]},{{76,185,243,308},10080 Sqrt[11]},{{112,162,230,308},10080 Sqrt[11]},{{49,207,269,287},10080 Sqrt[11]},{{37,222,270,283},10080 Sqrt[11]},{{70,192,270,280},10080 Sqrt[11]},{{142,162,230,278},10080 Sqrt[11]},{{126,184,250,252},10080 Sqrt[11]}}}

王守恩 · 发表于 6 天前

10楼是最好的。恰好有2个梯形属同一个最大面积！不是1个梯形(太多了)！！也不会有3个！！！
难得的一串数，2个不同图形属同一个最大面积, 这在四边形里面是独一的！

其它的，丢了！不可惜！譬如：面积=2025。随便拉一拉，就是15个。
{{a -> 7, b -> 75, c -> 85, d -> 47}, {a -> 15, b -> 27, c -> 123, d -> 135}, {a -> 15, b -> 45, c -> 75, d -> 75},
{a -> 31, b -> 45, c -> 53, d -> 59}, {a -> 33, b -> 45, c -> 51, d -> 57}, {a -> 51, b -> 25, c -> 65, d -> 111},
{a -> 57, b -> 27, c -> 45, d -> 93}, {a -> 79, b -> 15, c -> 113, d -> 191}, {a -> 117, b -> 15, c -> 39, d -> 153},
{a -> 125, b -> 15, c -> 25, d -> 145}, {a -> 131, b -> 15, c -> 17, d -> 139}, {a -> 205, b -> 9, c -> 41, d -> 245},
{a -> 219, b -> 9, c -> 15, d -> 231}, {a -> 399, b -> 5, c -> 13, d -> 411}, {a -> 673, b -> 3, c -> 5, d -> 677}}

Solve[{(b (d + a))/2 == 2025, 0 < a < d, 0 < b < c < 200, c^2 == b^2 + (d - a)^2, c - b < d - a < b + c}, {a, b, c, d}, Integers]

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要不来一个：周长=2025可以有几个梯形？4边=不同整数。我不会。求助了没人会做。OEIS没有这串数。

northwolves · 发表于 5 天前

周长=2025可以有几个梯形？4边=不同整数。
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85985676

王守恩 · 发表于 5 天前

接16楼。其它的，丢了！不可惜！譬如：面积=7！随便拉一拉，就是88个。算式简化了。

Solve[{(b (d + a))/2 == 7!, 0 < a < d, 0 < b < c < 300, c^2 == b^2 + (d - a)^2}, {a, b, c, d}, Integers]

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王守恩 · 发表于 5 天前

继续赞赏10楼！ 2个不同梯形同属一个最大面积。2个不同图形同属一个面积极值, 这是独一的！
周长={30,68,70,124,126,196,198,284,286,388,390,508,510,644,646,796,798,964,966,1148,1150,1348,1350,1564,1566,1796,1798,2044,2046,2308,2310,2588,2590,...}
a(1)=030=06+07+08+09=05+07+08+10,{{a -> 6, b -> 7, d -> 9}},
a(2)=068=14+17+18+19=13+17+18+20,{{a -> 14, b -> 17, d -> 19}},
a(3)=070=15+17+18+20=14+17+18+21,{{a -> 15, b -> 17, d -> 20}},
a(4)=124=27+31+32+34=26+31+32+35,{{a -> 27, b -> 31, d -> 34}},
a(5)=126=28+31+32+35=27+31+32+36,{{a -> 28, b -> 31, d -> 35}},
a(6)=196=44+49+50+53=43+49+50+54,{{a -> 44, b -> 49, d -> 53}},
a(7)=198=45+49+50+54=44+49+50+55,{{a -> 45, b -> 49, d -> 54}},
a(8)=284=65+71+72+76=64+71+72+77,{{a -> 65, b -> 71, d -> 76}},
a(9)=286=66+71+72+77=65+71+72+78,{{a -> 66, b -> 71, d -> 77}}

Table[Solve[{((d + a) Sqrt[((2 b + 1)^2 - (d - a)^2) ((d - a)^2 - 1)])/(4 (d - a)) == ((d + a) Sqrt[((2 b + 1)^2 - (d - a + 2)^2) ((d - a + 2)^2 - 1)])/(4 (d - a + 2)),
a + 2 b + 1 + d == 2 (n + 3) (n + 4 + Cos[n Pi]) - 2, 0 < a < b < d, n < d - a < 2 b + 1}, {a, b, d}, Integers], {n, 9}]

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加快速度！

Table[Solve[{((2 b - d + a + 1) (2 b + d - a + 1))/((2 b - d + a - 1) (2 b + d - a + 3)) == ((d - a + 3) (d - a)^2)/((d - a - 1) (d - a + 2)^2),
a + 2 b + d == (6 + 2 n + (21 + 14 n + 2 n^2) Cos[n Pi]) Cos[n Pi], 0 < a < b < d}, {a, b, d}, Integers], {n, 999}]

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northwolves · 发表于 4 天前

王守恩发表于 2024-12-28 17:19
继续赞赏10楼！ 2个不同梯形同属一个最大面积。2个不同图形同属一个面积极值, 这是独一的！
周长={30,68,70 ...

公式都有了，就不需要解方程了：

Table[cc=2(n+3)(n+4+(-1)^n) -2;
b=Floor[cc/4];
r=2+Ceiling[n/2];
{cc,{cc-3b-r-1,b,b+1,b+r},{cc-3b-r,b,b+1,b+r-1}},{n,30}]//MatrixForm

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30 {5,7,8,10} {6,7,8,9}
68 {13,17,18,20} {14,17,18,19}
70 {14,17,18,21} {15,17,18,20}
124 {26,31,32,35} {27,31,32,34}
126 {27,31,32,36} {28,31,32,35}
196 {43,49,50,54} {44,49,50,53}
198 {44,49,50,55} {45,49,50,54}
284 {64,71,72,77} {65,71,72,76}
286 {65,71,72,78} {66,71,72,77}
388 {89,97,98,104} {90,97,98,103}
390 {90,97,98,105} {91,97,98,104}
508 {118,127,128,135} {119,127,128,134}
510 {119,127,128,136} {120,127,128,135}
644 {151,161,162,170} {152,161,162,169}
646 {152,161,162,171} {153,161,162,170}
796 {188,199,200,209} {189,199,200,208}
798 {189,199,200,210} {190,199,200,209}
964 {229,241,242,252} {230,241,242,251}
966 {230,241,242,253} {231,241,242,252}
1148 {274,287,288,299} {275,287,288,298}
1150 {275,287,288,300} {276,287,288,299}
1348 {323,337,338,350} {324,337,338,349}
1350 {324,337,338,351} {325,337,338,350}
1564 {376,391,392,405} {377,391,392,404}
1566 {377,391,392,406} {378,391,392,405}
1796 {433,449,450,464} {434,449,450,463}
1798 {434,449,450,465} {435,449,450,464}
2044 {494,511,512,527} {495,511,512,526}
2046 {495,511,512,528} {496,511,512,527}
2308 {559,577,578,594} {560,577,578,593}

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[提问] 周长,面积相等的不同梯形

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