If the roots of $x^2-2\left(a+b\right)x+a^2+b^2+2c^2=0$  are equal then the relation between $a,b$  and c is

roots are equal so $\Delta =0$

$b^2=4.a.c$

${\left[-2\left(a+b\right)\right]}^2=4\times 1\times \left(a^2+b^2+2c^2\right)$

$4\left(a^2+b^2+2ab\right)=4\left(a^2+b^2+2c^2\right)$

$a^2+b^2+2ab=a^2+b^2+2c^2$

$2ab=2c^2$

$ab=c^2$