[yyabmuto] A set of triangles

Triangles inscribed in a circle, say clock face.  One vertex at 12.  Other two such that the center of the circle must be contained.

If vertices can only be at discrete points, e.g., clock hours, how many triangles are there (of course depends on whether center on edge is permitted)?  If unconstrained, then a large collection nice shapes.

Quadrilaterals?  Tetrahedra?