The. inertial migration of both spherical and oblate particles within an equilateral triangular channel is studied numerically. Our study primarily focuses on the effects of fluid inertia, quantified by the Reynolds number (Re) and particle size (β). Our observations reveal two distinct equilibrium positions: the corner equilibrium position (CEP) is situated along the angle bisector near the corner, while the face equilibrium position (FEP) is located on a segment of the line perpendicular from the triangle's center to one of its sides. Spherical particles with varying initial positions predominantly reach the FEP. For oblate particles initially positioned along the angle bisector with a specific orientation, meaning the particle's evolution axis is inside the plane bisecting the angle, they will migrate along the angle bisector to reach the CEP while rotating in the tumbling mode. Conversely, for particles with different initial orientations and positions, they will employ the log-rolling mode to reach the FEP. Notably, we identify a dual-stage particle migration process to the FEP, with trajectories converging to an equilibrium manifold, which bears a resemblance to the cross section of the channel. To further illustrate the transition between FEP and CEP under general initial conditions, except for those along the angle bisector, we construct a phase diagram in the (Re, β) parameter space. This transition is often triggered by the size of larger particles (as the FEP cannot accommodate them) or the influence of inertia for smaller particles. For the FEP, especially for medium- or small-size particles, we notice an initial outward movement of the FEP from the center of the cross section as Re increases, followed by a return towards the center. This behavior results from the interplay of three forces acting on the particle. This research holds potential implications for the design of microfluidic devices, offering insights into the behavior of particles within equilateral triangular channels.