{52,{{{6,9,16,21},36 Sqrt},{{4,10,18,20},36 Sqrt},{{5,10,18,19},36 Sqrt}}}
{58,{{{8,12,16,22},45 Sqrt},{{4,15,19,20},45 Sqrt},{{11,12,16,19},45 Sqrt}}}
{60,{{{10,14,16,20},120 Sqrt},{{12,13,15,20},120 Sqrt},{{12,14,16,18},120 Sqrt}}}
{108,{{{14,22,30,42},240 Sqrt},{{18,20,28,42},240 Sqrt},{{9,25,35,39},240 Sqrt},{{3,33,35,37},240 Sqrt}}},
{120,{{{15,29,31,45},120 Sqrt},{{18,26,34,42},120 Sqrt},{{20,26,34,40},120 Sqrt},{{28,29,31,32},120 Sqrt}}}
{184,{{{24,29,56,75},396 Sqrt},{{14,33,63,74},396 Sqrt},{{27,29,56,72},396 Sqrt},{{14,34,66,70},396 Sqrt},{{20,33,63,68},396 Sqrt}}}
{186,{{{23,42,52,69},336 Sqrt},{{13,52,56,65},336 Sqrt},{{21,44,58,63},336 Sqrt},{{25,44,58,59},336 Sqrt},{{29,43,56,58},336 Sqrt}}}}
{236,{{{13,57,75,91},720 Sqrt},{{10,60,76,90},720 Sqrt},{{18,52,76,90},720 Sqrt},{{31,45,71,89},720 Sqrt},{{34,45,71,86},720 Sqrt},{{38,50,72,76},720 Sqrt}}}
{264,{{{42,50,60,112},1320 Sqrt},{{22,62,70,110},1320 Sqrt},{{6,70,84,104},1320 Sqrt},{{32,49,83,100},1320 Sqrt},{{33,49,83,99},1320 Sqrt},{{60,62,70,72},1320 Sqrt}}}
{340,{{{42,74,86,138},2160 Sqrt},{{50,64,96,130},2160 Sqrt},{{35,72,108,125},2160 Sqrt},{{58,64,96,122},2160 Sqrt},{{8,99,113,120},2160 Sqrt},{{44,72,108,116},2160 Sqrt},{{5,109,111,115},2160 Sqrt}}}
{362,{{{25,81,101,155},540 Sqrt},{{27,73,109,153},540 Sqrt},{{37,63,114,148},540 Sqrt},{{38,63,119,142},540 Sqrt},{{41,63,119,139},540 Sqrt},{{64,73,109,116},540 Sqrt},{{76,81,101,104},540 Sqrt}}}
8个的：
{500,{{{85,108,120,187},360 Sqrt},{{60,124,136,180},360 Sqrt},{{90,103,127,180},360 Sqrt},{{96,102,128,174},360 Sqrt},{{68,116,144,172},360 Sqrt},{{85,108,137,170},360 Sqrt},{{96,109,135,160},360 Sqrt},{{85,116,144,155},360 Sqrt}}},
{636,{{{62,151,175,248},8400 Sqrt},{{93,125,175,243},8400 Sqrt},{{110,118,168,240},8400 Sqrt},{{66,140,196,234},8400 Sqrt},{{30,174,212,220},8400 Sqrt},{{21,197,199,219},8400 Sqrt},{{118,125,175,218},8400 Sqrt},{{94,140,196,206},8400 Sqrt}}},{760,{{{55,192,211,302},10710 Sqrt},{{40,204,220,296},10710 Sqrt},{{125,136,204,295},10710 Sqrt},{{70,163,240,287},10710 Sqrt},{{65,168,252,275},10710 Sqrt},{{140,142,210,268},10710 Sqrt},{{86,168,252,254},10710 Sqrt},{{107,163,240,250},10710 Sqrt}}},
{768,{{{104,172,180,312},2880 Sqrt},{{84,165,219,300},2880 Sqrt},{{104,152,226,286},2880 Sqrt},{{78,166,242,282},2880 Sqrt},{{24,220,260,264},2880 Sqrt},{{132,152,220,264},2880 Sqrt},{{104,166,242,256},2880 Sqrt},{{144,165,219,240},2880 Sqrt}}},{780,{{{120,174,186,300},5040 Sqrt},{{75,203,217,285},5040 Sqrt},{{138,156,204,282},5040 Sqrt},{{66,205,239,270},5040 Sqrt},{{150,156,204,270},5040 Sqrt},{{96,182,238,264},5040 Sqrt},{{102,183,240,255},5040 Sqrt},{{110,182,238,250},5040 Sqrt}}},
{800,{{{136,147,173,344},5040 Sqrt},{{80,177,203,340},5040 Sqrt},{{96,142,238,324},5040 Sqrt},{{37,181,259,323},5040 Sqrt},{{85,147,245,323},5040 Sqrt},{{12,196,289,303},5040 Sqrt},{{130,142,238,290},5040 Sqrt},{{112,153,255,280},5040 Sqrt}}},
{800,{{{57,181,239,323},2520 Sqrt},{{70,168,240,322},2520 Sqrt},{{64,171,245,320},2520 Sqrt},{{23,212,252,313},2520 Sqrt},{{120,140,228,312},2520 Sqrt},{{103,147,241,309},2520 Sqrt},{{40,184,280,296},2520 Sqrt},{{36,189,287,288},2520 Sqrt}}}
9个的：
{700,{{{130,132,148,290},13860 Sqrt},{{98,144,171,287},13860 Sqrt},{{107,129,186,278},13860 Sqrt},{{42,176,209,273},13860 Sqrt},{{95,134,206,265},13860 Sqrt},{{70,147,223,260},13860 Sqrt},{{20,198,222,260},13860 Sqrt},{{108,134,206,252},13860 Sqrt},{{91,147,223,239},13860 Sqrt}}},
{790,{{{110,160,201,319},11700 Sqrt},{{75,185,215,315},11700 Sqrt},{{135,143,197,315},11700 Sqrt},{{120,149,225,296},11700 Sqrt},{{95,160,240,295},11700 Sqrt},{{35,207,258,290},11700 Sqrt},{{122,156,234,278},11700 Sqrt},{{115,160,240,275},11700 Sqrt},{{170,185,215,220},11700 Sqrt}}}
{812,{{{120,178,186,328},10080 Sqrt},{{98,180,212,322},10080 Sqrt},{{76,185,243,308},10080 Sqrt},{{112,162,230,308},10080 Sqrt},{{49,207,269,287},10080 Sqrt},{{37,222,270,283},10080 Sqrt},{{70,192,270,280},10080 Sqrt},{{142,162,230,278},10080 Sqrt},{{126,184,250,252},10080 Sqrt}}}
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]
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。
周长=2025可以有几个梯形？4边=不同整数。
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85985676
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]
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)/(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) - 2, 0 < a < b < d, n < d - a < 2 b + 1}, {a, b, d}, Integers], {n, 9}]
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) Cos, 0 < a < b < d}, {a, b, d}, Integers], {n, 999}]
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;
r=2+Ceiling;
{cc,{cc-3b-r-1,b,b+1,b+r},{cc-3b-r,b,b+1,b+r-1}},{n,30}]//MatrixForm
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}