In
A Tangential Trapezoid, Jer describes a specific isosceles tangential trapezoid. I am going to generalize that idea.
How many isosceles tangential trapezoids exist where all their sides and diagonals are whole numbers and all those dimensions are at most 1000?
An example is a trapezoid with bases 6 and 20, lateral sides of 13, and diagonals of 17.
Bonus challenge: How many of these trapezoids have all dimensions at most 100,000?
clearvars,clc
ct=0;
for b=1:250
for t=b:250
h=sqrt(b*t); s=(t-b)/(2*sin(atan((t-b)/(2*h))));
if round(s)==sym(s)
bottomAngle=90+atand((t-b)/(2*h));
dsq= s^2+b^2-2*s*b*cosd(bottomAngle) ;
d=sqrt(dsq);
if round(d)==sym(d)
tot=t+b+2*(s+d);
if tot<=1000
fprintf('%5d %5s %5d %5s %8d
' ,b,sym(s),t,sym(d),round(tot));
ct=ct+1;
end
end
end
end
end
ct
finds 85 such trapezoids with totals at most 1000.
Caveats:
Testing for the diagonal or the side length being an integer, in the program, relies on Matlab's sym (i.e., symbolic) function recognizing a "close" approximation to an integer as being that integer. In the case of the square root result, I'd think it would be accurate; the case of the trig functions, perhaps less so. An example of Matlab's latitude in accepting approximation is its recognition of 1/3:
>> third=1/3
third =
0.333333333333333
>> sym(third)
ans =
1/3
>> 3*ans
ans =
1
>> ans==1
ans =
1 == 1
>> eval(ans)
ans =
logical
1
The evaluation of the total of the required dimensions is taken to be the top + bottom + twice the sum of the side length and the diagonal length, as there are two sides and two diagonals.
A given pair of bases, such as 6 and 20 (bottom and top) is counted only once, when the top is greater than the top, as it is the same trapezoid as when flipped so 20 is on bottom and 6 is on top.
Bases were considered only up to 250, as larger bases, together with sides and diagonals, would have totals exceeding 1000.
base side top diagonal total
4 5 6 7 34
6 13 20 17 86
8 10 12 14 68
8 25 42 31 162
10 17 24 23 114
10 41 72 49 262
12 15 18 21 102
12 26 40 34 172
12 61 110 71 386
14 37 60 47 242
14 85 156 97 534
16 20 24 28 136
16 50 84 62 324
16 113 210 127 706
18 39 60 51 258
18 65 112 79 418
20 25 30 35 170
20 34 48 46 228
20 82 144 98 524
22 101 180 119 642
24 30 36 42 204
24 52 80 68 344
24 75 126 93 486
24 122 220 142 772
28 29 30 41 198
28 35 42 49 238
28 74 120 94 484
30 51 72 69 342
30 65 100 85 430
30 123 216 147 786
32 40 48 56 272
32 100 168 124 648
36 45 54 63 306
36 53 70 73 358
36 78 120 102 516
36 130 224 158 836
40 50 60 70 340
40 68 96 92 456
40 125 210 155 810
42 65 88 89 438
42 91 140 119 602
42 111 180 141 726
44 55 66 77 374
44 85 126 113 566
48 60 72 84 408
48 89 130 119 594
48 104 160 136 688
50 85 120 115 570
52 65 78 91 442
52 125 198 161 822
54 117 180 153 774
56 58 60 82 396
56 70 84 98 476
56 148 240 188 968
60 75 90 105 510
60 102 144 138 684
60 130 200 170 860
60 149 238 191 978
64 80 96 112 544
66 73 80 103 498
66 143 220 187 946
68 85 102 119 578
70 119 168 161 798
72 90 108 126 612
72 106 140 146 716
76 95 114 133 646
78 109 140 151 738
80 100 120 140 680
80 136 192 184 912
84 87 90 123 594
84 105 126 147 714
84 130 176 178 876
88 110 132 154 748
90 97 104 137 662
92 115 138 161 782
96 120 144 168 816
100 125 150 175 850
104 130 156 182 884
108 135 162 189 918
112 116 120 164 792
112 140 168 196 952
116 145 174 203 986
120 137 154 193 934
132 146 160 206 996
140 145 150 205 990
Edited on December 16, 2022, 10:20 am
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Posted by Charlie
on 2022-12-16 10:17:50 |
computer solution (hope it's right--see caveats)
