The spin-1/2 triangular-lattice Heisenberg antiferromagnet with a-Type distortion is studied by the numerical-diagonalization method. We examine this model between the two cases, one is the undistorted triangular lattice and the other is the model on the honeycomb lattice with isolated spins. When the distortion is controlled, we find a nontrivial region where the ground states shows spontaneous magnetization
the magnitude increases gradually as the distortion is larger and is smaller than one third of the saturated magnetization.