  We investigate the phase behavior of a system of hard equilateral and
right-angled triangles in two dimensions using Monte Carlo simulations. Hard
equilateral triangles undergo a continuous isotropic-triatic liquid crystal
phase transition at packing fraction $\phi=0.7$. Similarly, hard right-angled
isosceles triangles exhibit a first-order phase transition from an isotropic
fluid phase to a rhombic liquid crystal phase with a coexistence region $\phi
\in \left[0.733,0.782\right]$. Both these liquid crystal phases undergo a
continuous phase transition to their respective close-packed crystal structures
at high pressures. Although the particles and their close-packed crystals are
both achiral, the solid phases of equilateral and right-angled triangles
exhibit spontaneous chiral symmetry breaking at sufficiently high packing
fractions. The colloidal triangles rotate either in clockwise or anti-clockwise
direction with respect to one of the lattice vectors for packing fractions
higher than $\phi_\chi$. As a consequence, these triangles spontaneously form a
regular lattice of left- or right-handed chiral holes which are surrounded by
six triangles in the case of equilateral triangles and four or eight triangles
for right-angled triangles. Moreover, our simulations show a spontaneous
entropy-driven demixing transition of the right- and left-handed "enantiomers".
