= Solution
Let $a$ denote the disease <allele>. Complete <penetrance> with <recessive inheritance> makes each affected child $aa$. The unaffected parents must therefore both be $Aa$ <genetic carriers>. Condition on these parental <genotypes> and the fully informative marker; the distinct paternal and maternal marker copies identify which parent's copies are shared. Average over the two equally likely marker-disease <haplotype> phases in each parent. This is the phase-symmetric model implicit in the requested <pedigree likelihood> identities.
For one parent, suppose first that both children receive the same marker copy. If that marker copy is on the parental <chromosome> carrying $a$, both children receive $a$ with <probability> $(1-\theta)^2$; in the opposite phase the <probability> is $\theta^2$. The phase-averaged <probability> is
$$
v=\frac{(1-\theta)^2+\theta^2}{2}=\frac12-\theta(1-\theta).
$$
If the children receive different marker copies, one disease-<allele> transmission is recombinant and the other is nonrecombinant in either phase. The <probability> that both receive $a$ is therefore
$$
u=\theta(1-\theta).
$$
Paternal and maternal transmissions are independent. Sharing zero marker copies means the different-copy case for both parents; sharing two means the same-copy case for both; sharing one means one of each. Thus the <recessive affected-sib-pair likelihood> is
$$
\boxed{\begin{aligned}
L_0&=\theta^2(1-\theta)^2=u^2,\\
L_1&=\frac{\theta(1-\theta)}2\bigl[(1-\theta)^2+\theta^2\bigr]=uv,\\
L_2&=\frac14\bigl[(1-\theta)^2+\theta^2\bigr]^2=v^2.
\end{aligned}}
$$
Here $L_j$ is the <probability> that both children are affected conditional on the marker sharing state $J=j$, not the <probability> of $J=j$ conditional on their being affected. Since $u,v\ge0$,
$$
\boxed{\sqrt{L_0L_2}=uv=L_1,\qquad \sqrt{L_0}+\sqrt{L_2}=u+v=\frac12.}
$$
At $\theta=1/2$, all three conditional disease <probabilities> equal $1/16$, as expected from two independent children of an $Aa\times Aa$ mating. At $\theta=0$, only the two-copy-sharing state can produce two affected children. Phase weights other than $1/2$ would require a different <likelihood function> and need not obey these identities.
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