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I need help in finding/solving these types of algebraic equations using Mathematica. here is the problem/code. I tried using 'With' command but no use. Thanks in advance.

If, a + b = c, then Sqrt[(a^4 + b^4 + c^4)/2] = (a^2 + b^2 + c^2)/2

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ForAll[{a, b, c}, a + b == c, 
  Sqrt[(a^4 + b^4 + c^4)/2] == (a^2 + b^2 + c^2)/2];
Resolve[%, Reals]

True

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another option could be

ClearAll[a, b, c];
lhs = Sqrt[(a^4 + b^4 + c^4)/2];
rhs = (a^2 + b^2 + c^2)/2;
Simplify[lhs - rhs, {a + b == c, Element[{b, c}, Reals]}]

(* 0 *)
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Using With works if both sides are squared first:

With[{c = a + b},
  (a^4 + b^4 + c^4)/2 == ((a^2 + b^2 + c^2)/2)^2] // Simplify

which evaluates to True.

Another way to verify this in Mathematica is:

(SymmetricReduction[#, {a, b, c}, {e1, e2, e3}] /. e1 -> 0 &
   [(a^4 + b^4 + c^4)/2 - ((a^2 + b^2 + c^2)/2)^2]) == {0, 0}

which evaluates to True.

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