First determination of the CP content of D→π+π−π0D→π+π−π0 and D→K+K−π0D→K+K−π0

Quantum-correlated ψ(3770)→DD¯ decays collected by the CLEO-c experiment are used to perform first measurements of F+F+, the fractional CP  -even content of the self-conjugate decays D→π+π−π0D→π+π−π0 and D→K+K−π0D→K+K−π0. Values of 0.968±0.017±0.0060.968±0.017±0.006 and 0.731±0.058±0.0210.731±0.058±0.021 are obtained for π+π−π0π+π−π0 and K+K−π0K+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∓B∓→DK∓ decays. The high CP  -even content of D→π+π−π0D→π+π−π0, in particular, makes this a promising mode for improving the precision on γ.