= Solution
For an <ideal gas> of fixed composition, the central <temperature> satisfies $T_c\propto p_c/\rho_c=K\rho_c^{1/n}$. Keeping $T_c$ fixed therefore requires $K\propto\rho_c^{-1/n}$, rather than keeping the <polytropic constant> fixed. Substitution in the <Lane-Emden equation> scalings gives
$$
M\propto\rho_c^{(3/n-1)/2}\rho_c^{-3/(2n)}=\rho_c^{-1/2},\qquad
R\propto\rho_c^{(1/n-1)/2}\rho_c^{-1/(2n)}=\rho_c^{-1/2}.
$$
Consequently the <homologous polytropes at fixed central temperature> obey
$$
\boxed{M\propto R.}
$$
The <polytropic index> and mean particle <mass> must be common to this homologous sequence: varying either changes the dimensionless structural coefficient. The fixed-central-temperature assumption is the idealization being used here, rather than a universal property of real <stars>.
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