Given a rectangle $A$ and $n$ triangles $\{B_1,B_2,...,B_n\}$, I put the triangles inside $A$, at least one vertex of each triangle is not outside $A$ (inside $A$ or on the edge of $A$). So that A is split into $m$ polygons $\{C_1,C_2,...,C_m\}$ without overlapping.
How should I put $\{B_1,B_2,...,B_n\}$ in order to maximum $m$?
How should I put $\{B_1,B_2,...,B_n\}$ in order to minimize the variance of the areas of polygons $\{C_1,C_2,...,C_m\}$? namely to split the rectangle into polygons as uniformly as possible?