The triangle 5,12,13 has an area A=30 and a perimeter P=30, so A/P is 1.
The triangle 9,75,78 has an area A=324 and a perimeter P=162, so A/P is 2.
Find the smallest and largest integer-sided triangles where A/P is 10.
clc,clearvars
for a=1:100000
for b=1:a
for c=a-b+1:b
s=(a+b+c)/2;
p=2*s;
A=sqrt(s*(s-a)*(s-b)*(s-c));
if A/p==10
fprintf('%5d %5d %5d %16.14f %5d %16.14f\n',[a b c A p A/p])
end
end
end
end
The smallest is 75, 70, 65.
The abbreviated findings:
sides Area perim. ratio
75 70 65 2100.00000000000000 210 10.00000000000000
78 78 60 2160.00000000000000 216 10.00000000000000
87 74 61 2220.00000000000000 222 10.00000000000000
89 82 57 2280.00000000000000 228 10.00000000000000
97 72 65 2340.00000000000000 234 10.00000000000000
100 80 60 2400.00000000000000 240 10.00000000000000
106 90 56 2520.00000000000000 252 10.00000000000000
110 102 52 2640.00000000000000 264 10.00000000000000
116 82 66 2640.00000000000000 264 10.00000000000000
120 75 75 2700.00000000000000 270 10.00000000000000
123 122 49 2940.00000000000000 294 10.00000000000000
. . .
10504 10302 206 210120.00000000000000 21012 10.00000000000000
10601 10504 105 212100.00000000000000 21210 10.00000000000000
10841 10426 417 216840.00000000000000 21684 10.00000000000000
11175 11043 138 223560.00000000000000 22356 10.00000000000000
14667 14606 73 293460.00000000000000 29346 10.00000000000000
16850 16441 411 337020.00000000000000 33702 10.00000000000000
20602 20402 204 412080.00000000000000 41208 10.00000000000000
20858 20451 409 417180.00000000000000 41718 10.00000000000000
27534 27403 137 550740.00000000000000 55074 10.00000000000000
32885 32481 406 657720.00000000000000 65772 10.00000000000000
I can't tell if there are more beyond this. The program was stopped manually at a=38448 (a being the largest side of the triangle).
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Posted by Charlie
on 2024-01-04 08:56:35 |
partial solution
