If the roots of $\left({\mathrm{a}}^2+{\mathrm{b}}^2\right){\mathrm{x}}^2-2\mathrm{b}(\mathrm{a}+\mathrm{c})\mathrm{x}+\left({\mathrm{b}}^2+{\mathrm{c}}^2\right)=0$ are real and equal then  $a,b,c$are in

The given quadratic equation is $\left({\mathrm{a}}^2+{\mathrm{b}}^2\right){\mathrm{x}}^2-2\mathrm{b}(\mathrm{a}+\mathrm{c})\mathrm{x}+\left({\mathrm{b}}^2+{\mathrm{c}}^2\right)=0$

Given that the roots are real and equal

If the roots of the quadratic equation $a{x}^2+bx+c=0$ are real and equal then its discriminant value is  zero

It implies that $b^2-4ac=0$

Hence, 

    $\begin{array}{l}4{\mathrm{b}}^2(\mathrm{a}+\mathrm{c}{)}^2-4\cdot \left({\mathrm{a}}^2+{\mathrm{b}}^2\right)\left({\mathrm{b}}^2+{\mathrm{c}}^2\right)=0\end{array}$

Simplifying 

$\begin{array}{c}{\mathrm{a}}^2{\mathrm{b}}^2+2{\mathrm{acb}}^2+{\mathrm{b}}^2{\mathrm{c}}^2-{\mathrm{a}}^2{\mathrm{b}}^2-{\mathrm{a}}^2{\mathrm{c}}^2-{\mathrm{b}}^4-{\mathrm{b}}^2{\mathrm{c}}^2=0\\ b^4-2ac{b}^2+{\left(ac\right)}^2=0\\ {\left({\mathrm{b}}^2-\mathrm{ac}\right)}^2=0\\ b^2=ac\end{array}$

Therefore, $a,b,c$ are in $\boxed{Geometric progression}$