  \noindent The simultaneous partition problems are classical problems of the
combinatorial geometry which have the natural flavor of the equivariant
topology. The $k$-fan partition problems have attracted a lot of attention
\cite{Aki2000}, \cite{BaMa2001}, \cite{BaMa2002} and forced some hard concrete
combinatorial calculations in the equivariant cohomology \cite% {Bl-Vr-Ziv}.
These problems can be reduced, by a beautiful scheme of \cite% {BaMa2001}, to a
\textquotedblright typical\textquotedblright question of the existence of a
$\mathbb{D}_{2n}$ equivariant map $f:V_{2}(\mathbb{R}% ^{3})\to W_{n}-\cup
\mathcal{A}(\alpha)$, where $V_{2}(\mathbb{R}% ^{3})$ is the space of all
orthonormal 2-frames in $\mathbb{R}^{3}$ and $% W_{n}-\cup \mathcal{A}(\alpha)$
is the complement of the appropriate arrangement. We introduce the
\textit{target extension scheme} which allow us to use the equivariant
obstruction theory as a tool for proving that: for every two proper measures on
the sphere $S^{2}$, and any $\alpha =(a,a+b,b)\in \mathbb{R}_{>0}^{3}$, there
exists an $\alpha $-partition of theses measures by a 3-fan.
  \noindent The significance of these results, among other, is that, beside
negative results \cite{Bl-Vr-Ziv}, the equivariant obstruction theory can pull
off some positive results, which were not attained by other means.
