2026/051) Show that square of sum of 3 squares can be expressed as sum of 3 squares

 We need to show that $(a^2+b^2+ c^2)^2$ can be expressed as sum of 3 squares.

Let for simplicity 

$x= a^2+b^2\cdots(1)$ 

and 

$y=c^2\cdots(2)$

So we have

$(a^2+b^2 + c^2)^2 = (x+y)^2$ from (1) and (2)

$=(x-y)^2 + 4xy$ as $(x+y)^2 - (x-y)^2 = 4xy$ 

$=  (a^2+b^2-c^2)^2 + 4(a^2+b^2)c^2$ putting back values

$=  (a^2+b^2-c^2)^2 + 4a^2c^+4b^2c^2$

$=  (a^2+b^2-c^2)^2 + (2ac)^2 + (2bc)^2$

Hence proved