## estudier 3 years ago Show that area of Pythagorean triangle x,y,z cannot be square (U can use that a^4-b^4 = c^2 has no positive solution)

1. AbhimanyuPudi

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

2. AbhimanyuPudi

Its not generalised though!!

3. estudier

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

4. AbhimanyuPudi

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

5. experimentX

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6. experimentX

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7. estudier

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

8. estudier

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

9. estudier

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

10. experimentX

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11. estudier

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

12. experimentX

oh!!

13. estudier

:-)

14. experimentX

that would yield extra non dependent solution.

15. experimentX

*equation (solution)

16. estudier

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

17. estudier

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

18. experimentX

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

19. estudier

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

20. experimentX

yeah i see the trick!!

21. estudier

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

22. experimentX

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

23. estudier

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

24. experimentX

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

25. experimentX

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

26. estudier

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