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Math Help - 4 equations

  1. #1
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    4 equations

    4 equations, 4 unknowns:

    u(r^2 - u^2) / (r^2 + u^2) = 156 [1]
    v(r^2 - v^2) / (r^2 + v^2) = 96 [2]
    w(r^2 - w^2) / (r^2 + w^2) = 63 [3]
    (315uvw + 24336vw + 9216uw + 3969uv) / (2r) = 943488 [4]

    Who can solve that mess?
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  2. #2
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    Re: 4 equations

    Did you notice that all the integers are divisible by three? From equation four, we know that r is also divisible by three.
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  3. #3
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    Re: 4 equations

    Quote Originally Posted by zhandele View Post
    ...From equation four, we know that r is also divisible by three.
    Not so. I brute forced this one: r = 520. And u=260, v=104, w=65.

    I'm trying to do this by solving instead of brute force.
    Last edited by Wilmer; January 4th 2013 at 07:27 PM.
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  4. #4
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    Re: 4 equations

    Hi Wilmer, can you give us some clue as to background/origin ?
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  5. #5
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    Re: 4 equations

    Thanks Bob. Best I can do diagramwise(!):
    Code:
                  C
    
    
    
    
    
            D                     E
                  U          V                    
                       M
                       
                       W
                       
    B                  F                        A
    Acute triangle ABC: M is circumcenter.
    U, V and W are the incenters of triangles BCM, ACM and ABM respectively:
    and DM, EM and FM are the perpendicular heights.

    NOT GIVENS: a = BC = 624, b = AC = 960, c = AB = 1008, r = 520 = AM=BM=CM.
    NOT GIVENS: u = UM = 260, v = VM = 104, w = WM = 65.
    GIVENS: d = DU = 156, e = EV = 96, f = FW = 63.

    Work to set up the 4 equations:

    a = 2dr / u , b = 2er / b , c = 2fr / w

    from triangleBCM: u(r^2 - u^2) / (r^2 + u^2) = d [1]
    from triangleACM: v(r^2 - v^2) / (r^2 + v^2) = e [2]
    from triangleABM: w(r^2 - w^2) / (r^2 + w^2) = f [3]

    area(BCM + ACM + ABM) = areaABC; leads to :
    [uvw(d + e + f) + vwd^2 + uwe^2 + uvf^2] / (2r) = def [4]

    Inserting the givens gives us:

    u(r^2 - u^2) / (r^2 + u^2) = 156 [1]
    v(r^2 - v^2) / (r^2 + v^2) = 96 [2]
    w(r^2 - w^2) / (r^2 + w^2) = 63 [3]
    (315uvw + 24336vw + 9216uw + 3969uv) / (2r) = 943488 [4]

    I'm simply curious as to the possibility of solving these 4 simultaneous equations.
    Last edited by Wilmer; January 5th 2013 at 08:56 AM.
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