Let $\phi_1$ and $\phi_2$ two homotheties with centers $O_1$ and $O_2$ respectively. Their composition $\phi=\phi_1 \circ \phi_2$ is a translation if $\mu_1\mu_2=1$ (where $\mu_1$ and $\mu_2$ are the ratios of each homothety) or another homothety with ratio $\mu_1 \mu_2$ if $\mu_1\mu_2\neq 1$.
I would like to determine the barycentric coordinates of the center of the later homothety, which I call $O_\phi$, with respect to $\{O_1,O_2\}$.
I can see that if $O_1=O_2$, then $O_\phi=O_1$ (and then $\phi_1\phi_2=\phi_2\phi_1$), whereas if $O_1\neq O_2$ then $O_\phi$ would be situated somewhere on the line passing through $O_1$ and $O_2$, so the question makes sense to me. I can also see that for an arbitrary point $P$, $\phi_1(P)=O_1+\mu_1 \overrightarrow{O_1P}$ and $\phi_2(\phi_1(P))=O_2+\mu_2 \overrightarrow{O_2\phi_1(P)}$. But I’m not sure how to carry on from there.
I'm assuming $\phi=\phi_1\circ\phi_2$ and $P\to P''=\phi_1(P')=\phi_1(\phi_2(P))$. $$ P'=(1-\mu_2)O_2+\mu_2P; $$ $$ \begin{align} P''&=(1-\mu_1)O_1+\mu_1P'\\ &=(1-\mu_1)O_1+\mu_1\big((1-\mu_2)O_2+\mu_2P\big)\\ &=(1-\mu_1)O_1+\mu_1(1-\mu_2)O_2+\mu_1\mu_2P\\ &=(1-\mu_1\mu_2){(1-\mu_1)O_1+\mu_1(1-\mu_2)O_2\over1-\mu_1\mu_2} +\mu_1\mu_2P. \end{align} $$ Hence the composed homothety has ratio $\mu_1\mu_2$ and center: $$ O_{12}={(1-\mu_1)O_1+\mu_1(1-\mu_2)O_2\over1-\mu_1\mu_2}. $$