Authors’ abstract: We prove that for any absolute continuous Borel probability measure \(\mu\) on the sphere \(S^2\) and any \(t\in[0,\frac14]\) there exist four great semi-circles \(\ell_1,\ldots,\ell_4\) emanating from a point \(x\in S^2\) into four angular sectors \(\sigma_1,\ldots,\sigma_4\), counter clockwise oriented, such that \(\mu(\sigma_1)=\mu(\sigma_4)=t\), \(\mu(\sigma_2)=\mu(\sigma_3)=\frac14-t\), and \(\operatorname{area}(\sigma_1)=\operatorname{area}(\sigma_4)\), \(\operatorname{area}(\sigma_2)=\operatorname{area}(\sigma_3)\).