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We further study the "complementary" Ansatz, Tr(M_\nu)=0, for a prediagonal light Majorana type neutrino mass matrix. Previously, this was studied for the CP conserving case and the case where the two Majorana type CP violating phases were present but the Dirac type CP violating phase was neglected. Here we employ a simple geometric algorithm which enables us to "solve" the Ansatz including all three CP violating phases. Specifically, given the known neutrino oscillation data and an assumed two parameter (the third neutrino mass m_3 and the Dirac CP phase \delta) family of inputs we predict the neutrino masses and Majorana CP phases. Despite the two parameter ambiguity, interesting statements emerge. There is a characteristic pattern of interconnected masses and CP phases. For large m_3 the three neutrinos are approximately degenerate. The only possibility for a mass hierarchy is to have m_3 smaller than the other two. A hierarchy with m_3 largest is not allowed. Small CP violation is possible only near two special values of m_3. Also, the neutrinoless double beta decay parameter is approximately bounded as 0.020 eV <|m_{ee}|

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