Quantum-correlated psi(3770)->DDbar decays collected by the CLEO-c experiment are used to perform first measurements of F+, the fractional CP-even content of the self-conjugate decays D->pi+pi-pi0 and D->K+K-pi0. Values of 0.968 +/- 0.017 +/- 0.006 and 0.731 +/- 0.058 +/- 0.021 are obtained for pi+pi-pi0 and K+K-pi0, respectively. It is demonstrated how modes of this sort can be cleanly included in measurements of the unitarity triangle angle gamma using B+/- -> D K+/- decays. The high CP-even content of D -> pi+pi-pi0, in particular, makes this a promising mode for improving the precision on gamma.
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