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Title: Re: Heronian Triangle Post by Eigenray on Apr 26th, 2003, 7:16pm Well, I get that[hide] a2b = (b-c)(b+c)2[/hide], and that [hide] 4b2 - (b+c)2 is a perfect square[/hide]. Other than that I got nothing. |
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Title: Re: Heronian Triangle Post by Eigenray on Apr 30th, 2003, 5:39pm Why, it's [hide]985320[/hide], 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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