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Show that area of Pythagorean triangle x,y,z cannot be square
(U can use that a^4b^4 = c^2 has no positive solution)



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AbhimanyuPudi
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the base and height of pythagorean triangle are 3d and 4d where d is a constant
Area = 1/2 x 3d x 4d = 3d x 2d = 6d^2 this is never a square

AbhimanyuPudi
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Its not generalised though!!

estudier
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It's a start...
Let's say we have x^2 + y^2 = z^2 and xy = 2n^2 (triangle area)

AbhimanyuPudi
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Okay so we take it as hypothesis and prove it a contradiction..I'll try

experimentX
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dw:1350135906515:dw

experimentX
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dw:1350136054246:dw

estudier
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Promising...how will you get a quartic so as to use the given?

estudier
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You need to multiply the first line of your first post with something...

estudier
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Writing your first line in full:
(x+y)^2 = x^2 + y^2 +2xy = z^2 + 4n^2

experimentX
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dw:1350136543480:dw

estudier
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(x+y)^2 = x^2 + y^2 +2xy = z^2 + 4n^2 Your first line
(xy)^2 = x^2 + y^2 2xy = z^2 4n^2

experimentX
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oh!!

estudier
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:)

experimentX
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that would yield extra non dependent solution.

experimentX
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*equation (solution)

estudier
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We, are just trying to get a contradiction....

estudier
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We are assuming a triangle that is square....

experimentX
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yeah ... three variables 3 non dependent equations ... it's possible to deduce from that.

estudier
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If u multiply those two above, u get
(x^2y^2)^2 = z^4  (2n)^4
which contradicts the given
so no Pythagorean triangle area is square

experimentX
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yeah i see the trick!!

estudier
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U almost had it, a bit more time, u would have had it, I think....

experimentX
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well .. I'm not too good with algebraic manipulations.

estudier
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I shall put up a few more while it's stil quiet....

experimentX
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I usually takes quite lot more than necessary steps to do things.

experimentX
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I might go offline for 2hrs ... well I would enjoy if you tag me in few interesting problems if you have.

estudier
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