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

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) (Show Source): You can put this solution on YOUR website!

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

Write the last factor second:

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

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) ≟ 4a�b�

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

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

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

      [(a�+2ab+b�)-c�][-(b�-2ba+a�)+c�] ≟ 4a�b�

          [a�+2ab+b�-c�][-b�+2ba-a�+c�] ≟ 4a�b�

Substitute a�+b� for c�

[a�+2ab+b�-(a�+b�)][-b�+2ba-a�+(a�+b�)] ≟ 4a�b�

    [a�+2ab+b�-a�-b�][-b�+2ba-a�+a�+b�] ≟ 4a�b�

           [b�+2bc+c�-a�][a�-b�+2bc-c�] ≟ 4a�b�

Substitute a�+b� for each c�

 [b�+2bc+(a�+b�)-a�][a�-b�+2bc-(a�+b�)] ≟ 4a�b�

     [b�+2bc+a�+b�-a�][a�-b�+2bc-a�-b�] ≟ 4a�b�

                             [2ab][2ba] ≟ 4a�b�

                                  4a�b� ≡ 4a�b�

Edwin

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