Using an e(+)e(-) annihilation data sample corresponding to an integrated luminosity of 2.93 fb(-1) collected at the center-of-mass energy of 3.773 GeV with the BESIII detector, we measure the absolute branching fractions of D+ -> eta eta pi(+), D+ -> eta pi(+pi 0), and D-0 ->eta pi(+) pi(-) to be (2.96 +/- 0.24 +/- 0.10+x10(-3), (2.23 +/- 0.15 +/- 0.10)x10(-3), and (1.20 +/- 0.07 +/- 0.04) x 10(-3), respectively, where the first uncertainties are statistical and the second ones are systematic. The D+ -> eta eta pi(+) decay is observed for the first time, and the branching fractions of D+(0) -> eta pi(+) pi(0(-)) are measured with much improved precision. In addition we test for CP asymmetries in the separated charge-conjugate branching fractions; no evidence of CP violation is found.
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