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Math Help - One for you, WonderBoy

  1. #1
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    One for you, WonderBoy

    Right triangle: sides 135-352-377
    Isosceles triangle: sides 132-366-366
    Both have perimeter = 864 and area = 23760

    Find another case where a right triangle and an isosceles triangle have the
    same area and perimeter; these rules apply, of course:
    - all sides are integers
    - new case is primitive (not a multiple of above)
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  2. #2
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    Working on it Wilmer

    Without a computer, just a calculator. Give me a couple of days.
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  3. #3
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    Quote Originally Posted by wonderboy1953 View Post
    Without a computer, just a calculator. Give me a couple of days.
    Use computer if you wish.
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    This was a toughie

    Tried applying Goldbach's Conjecture. No success.

    Next I flipped Riemann's Conjecture. No luck here.

    I next tried a backward FLT with Selmer Groups and a little Galois's Group mixed in. Nothing doing here either.

    I guess we'll have to settle for Right Triangle: 270, 704, 754 and
    Isosceles Triangle: 264, 732, 732 with...

    perimeter of 1728 and area equaling 95,040.
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  5. #5
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    Quote Originally Posted by wonderboy1953 View Post
    I guess we'll have to settle for Right Triangle: 270, 704, 754 and
    Isosceles Triangle: 264, 732, 732 with...
    Wrong answer;those not primitive.
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    Back to the drawing board

    I'll keep trying.
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    If this really requires a computer

    Then I will decline this challenge until I think of a shortcut or can run a program on a computer to solve this problem.
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    Fyi

    I don't believe there is another primitive case.

    I discovered this in 2003, and sent it to MathWorld; they show it here:
    Heronian Triangle -- from Wolfram MathWorld
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    Very nice. So, can you prove that no other primitive case exists?

    In my naivete I was trying to take a "constrained optimization" approach to this problem. It of course didn't work, but I still am not totally clear as to why it didn't work.
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  10. #10
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    Quote Originally Posted by Wilmer View Post
    I don't believe there is another primitive case.

    I discovered this in 2003, and sent it to MathWorld; they show it here:
    Heronian Triangle -- from Wolfram MathWorld
    Did you run the lengths up to 400,000 on a computer (and how long did that take)?
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    Wilmer

    Did you know that you can only post problems you already know the answer to in this section? "I don't believe there is another primitive case."
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    YES, I know!
    However, this one HAS an answer: "it is not known..."
    A bit like the Euler Brick:
    Euler Brick -- from Wolfram MathWorld

    Perhaps Mr Fantastic can issue a statement as to
    "it is not known" being a valid solution HERE.
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  13. #13
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    Quote Originally Posted by Danny View Post
    Did you run the lengths up to 400,000 on a computer (and how long did that take)?
    Yes I did. How long? Forgot!
    I just did a run getting ALL right triangles where short leg < 100000:
    there are 1,521,629 triangles; took 12 minutes 7 seconds (included
    was the checking of each to see if it met the criteria).

    I hit on this quite accidentally; like, tried it "just to kill time"!
    I was surprised that MathWorld "accepted" it.

    My computer "search" is simple enough:
    get right triangle sides a,b,c (c being hypotenuse, of course)

    I use this triangle as half the isosceles triangle.

    p = 2(a + c) ; k = 2ab

    I then check if there is a right triangle d,e,f (d<e, f the hypotenuse)
    that meets the "same perimeter-area" and integer condition:
    d,e = [p^2 + 2k +- SQRT(p^4 - 12p^2 k + 4k^2)] / (4p)
    (d being the -, e being the +)

    So ONLY 2 variables are looped: a and b.
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