  Using the bond-propagation algorithm, we study the finite-size behavior of
the critical two-dimensional Ising model on a finite triangular lattice with
free boundaries in five shapes: triangle, rhombus, trapezoid, hexagon and
rectangle. The critical free energy, internal energy and specific heat are
calculated. The accuracy of the free energy reaches $10^{-26}$. Based on
accurate data on several finite systems with linear size up to N=2000, we
extract the bulk, surface and corner parts of the free energy, internal energy
and specific heat accurately. We confirm the conformal field theory prediction
of the corner free energy to be universal and find logarithmic corrections in
higher order terms in the critical free energy for the rhombus, trapezoid, and
hexagon shaped systems, which are absent for the triangle and rectangle shaped
systems. The logarithmic edge corrections due to edges parallel or
perpendicular to the bond directions in the internal energy are found to be
identical, while the logarithmic edge corrections due to corresponding edges in
the free energy and the specific heat are different. The corner internal energy
and corner specific heat for angles $\pi/3$, $\pi/2$ and $2\pi/3$ are obtained,
as well as higher order corrections. Comparing with the corner internal energy
and corner specific heat previously found on a rectangle of the square lattice
(Phys. Rev. E. 86 041149 (2012)), we conclude that the corner internal energy
and corner specific heat for the rectangle shape are not universal.
