If ${\mathrm{a}}^2+{\mathrm{b}}^2,\mathrm{ab}+\mathrm{bc} \mathrm{and} {\mathrm{b}}^2+{\mathrm{c}}^2$ are in G.P., then a, b, c are in

$\begin{array}{l}{\mathrm{a}}^2+{\mathrm{b}}^2,\mathrm{ab}+\mathrm{bc},{\mathrm{b}}^2+{\mathrm{c}}^2\text{ are in G.P. }\\ \Rightarrow (\mathrm{ab}+\mathrm{bc}{)}^2=\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)\left({\mathrm{b}}^2+{\mathrm{c}}^2\right)\\ \text{ or }{\mathrm{a}}^2{\mathrm{b}}^2+{\mathrm{b}}^2{\mathrm{c}}^2+2{\mathrm{ab}}^2\mathrm{c}={\mathrm{a}}^2{\mathrm{b}}^2+{\mathrm{a}}^2{\mathrm{c}}^2+{\mathrm{b}}^2{\mathrm{c}}^2+{\mathrm{b}}^4\\ \text{ or }{\mathrm{b}}^4+{\mathrm{a}}^2{\mathrm{c}}^2-2{\mathrm{ab}}^2\mathrm{c}=0\\ \text{ or }{\left({\mathrm{b}}^2-\mathrm{ac}\right)}^2=0\\ \text{ or }{\mathrm{b}}^2=\mathrm{ac}\end{array}$
Hence, a, b, and c are in G.P.