If $\alpha ,\beta ,\gamma$  are the geometric means between $ca,ab;ab,bc;bc,ca$  respectively where $a,b,c$  are in A.P. then ${\alpha }^2,{\beta }^2,{\gamma }^2$  are in:

By hypothesis, ${\alpha }^2=a^{2}bc,{\beta }^2=b^{2}ca,{\gamma }^2=c^{2}ab$ and $2b=a+c$

$\Rightarrow$           $2a{b}^{2}c=a^{2}bc+ab{c}^2$

$\Rightarrow$          $2{\beta }^2={\alpha }^2+{\gamma }^2$

Hence, ${\alpha }^2,{\beta }^2,{\gamma }^2$  are in A.P