SOLUTION: if a^2+b^2=c^2 than prove that (a+b+c)(b+c-a)(c+a-b)(a+b-c)=4a^2b^2

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Question 789866: if a^2+b^2=c^2 than prove that (a+b+c)(b+c-a)(c+a-b)(a+b-c)=4a^2b^2
Answer by Edwin McCravy(20063) About Me  (Show Source):
You can put this solution on YOUR website!
           (a+b+c)(b+c-a)(c+a-b)(a+b-c) ≟ 4a²b²

Write the last factor second:

           (a+b+c)(a+b-c)(b+c-a)(c+a-b) ≟ 4a²b²

Rearrange and regroup each factor:

(a+b+c) = [b+c+a] = [(b+c)+a]
(a+b-c) = [a-c+b] = [a+b-c] = [a+(b-c)]
(b+c-a) = [b+c-a] = [(b+c)-a] 
(c+a-b) = [a-b+c] = [a-(b-c)]


           (a+b+c)(a+b-c)(b+c-a)(c+a-b) ≟ 4a²b²

      [(a+b)+c][(a+b)-c][b-a+c][-b+a+c] ≟ 4a²b²

  [(a+b)+c][(a+b)-c][(b-a)+c][-(b-a)+c] ≟ 4a²b²

                [(a+b)²-c²][-(b-a)²+c²] ≟ 4a²b²

      [(a²+2ab+b²)-c²][-(b²-2ba+a²)+c²] ≟ 4a²b²

          [a²+2ab+b²-c²][-b²+2ba-a²+c²] ≟ 4a²b²

Substitute a²+b² for c²

[a²+2ab+b²-(a²+b²)][-b²+2ba-a²+(a²+b²)] ≟ 4a²b²

    [a²+2ab+b²-a²-b²][-b²+2ba-a²+a²+b²] ≟ 4a²b²

           [b²+2bc+c²-a²][a²-b²+2bc-c²] ≟ 4a²b²

Substitute a²+b² for each c²

 [b²+2bc+(a²+b²)-a²][a²-b²+2bc-(a²+b²)] ≟ 4a²b²

     [b²+2bc+a²+b²-a²][a²-b²+2bc-a²-b²] ≟ 4a²b²

                             [2ab][2ba] ≟ 4a²b²

                                  4a²b² ≡ 4a²b²

Edwin