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Topic: Re: Heronian Triangle (Read 914 times) |
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Eigenray
wu::riddles Moderator
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Posts: 1948
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Re: Heronian Triangle
« on: Apr 26th, 2003, 7:16pm » |
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Well, I get that
a2b = (b-c)(b+c)2,
and that
4b2 - (b+c)2 is a perfect square.
Other than that I got nothing.
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Eigenray
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 1948
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Re: Heronian Triangle
« Reply #1 on: Apr 30th, 2003, 5:39pm » |
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Why, it's 985320, of course!
Is there a way to do it short of brute force?
Basically what I did was:
I factored each c < 10000, and then looped through all b > c such that b | c3. I first checked if
a=(b+c)sqrt((b-c)/b)
was an integer, and if it was, I checked whether
K=c(b+c)sqrt(3b2-2cb-c2)/(4b) was also.
Here are the solutions for c < 5000, sorted by area:
K, a, b, c
985320, 2380, 2197, 897
1275120, 5152, 4913, 561
3941280, 4760, 4394, 1794
5100480, 10304, 9826, 1122
8867880, 7140, 6591, 2691
11476080, 15456, 14739, 1683
15765120, 9520, 8788, 3588
20401920, 20608, 19652, 2244
24633000, 11900, 10985, 4485
31878000, 25760, 24565, 2805
45904320, 30912, 29478, 3366
62480880, 36064, 34391, 3927
81607680, 41216, 39304, 4488
(This took about 1 second to run. Testing each b < c3/2 to see if b | c3, rather than running through the allowable prime factorizations, takes about 4 minutes.)
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| « Last Edit: Apr 30th, 2003, 8:00pm by Eigenray » |
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