First determination of the CP content of D→π⁺Ï€^-Ï€⁰ and D→K⁺K^-Ï€⁰

Quantum-correlated ψ(3770)→DD 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→π+π-π0 and D→K+K-π0. Values of 0.968±0.017±0.006 and 0.731±0.058±0.021 are obtained for π+π-π0 and K+K-π0, respectively. It is demonstrated how modes of this sort can be cleanly included in measurements of the unitarity triangle angle γ using B∓→DK∓ decays. The high CP-even content of D→π+π-π0, in particular, makes this a promising mode for improving the precision on γ.