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November 2nd, 2007, 08:47 PM   #1
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How many triangles?

How many distinct triangles exist such that one side is 6 and the other two sides are integers and the perimeter numerically equal to the area.
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November 3rd, 2007, 12:07 PM   #2
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Let a and b (which are any positive integers) the other lengths of the triangle. Using Heron's theorem,
a+b+6=
sqrt((a+b+6 / 2)(a+b+6 / 2 - a)(a+b+6 / 2 - b)(a+b+6 / 2 - c)).
Let a+b+6 / 2 = k. Then, 2k=sqrt(k(k-a)(k-b)(k-c)).
4k^2 = k(k-a)(k-b)(k-6)
4k = (k-a)(k-b)(k-6) = k^3 - (a+b+6)k^2 + (ab+6b+6a)k - 6ab
0 = k^3 - (a+b+6)k^2 + (ab+6b+6a-4)k - 6ab
Use Cardano's Method of cubic equations to solve for k.
After using Cardano's Method, you have to use some more algebraic factoring and substitution, and little theories, and you should arrive with the possible number of distinct triangles for sides a and b, where a and b are positive integers.
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November 6th, 2007, 06:59 AM   #3
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I suspect, but haven't proved yet, that there is only one such triangle (with sides 6, 8 and 10).
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