Perfect Mondrian Triangle Dissections

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In the Mondrian Art Problem, a square is divided into non-congruent integer-sided rectangles so that the largest area and smallest area are as close as possible.

A lattice square can be divided into non-congruent lattice-vertex triangles so that all areas are identical. Pick's theorem, Area = 2 Interior + Boundary/2 - 1, is handy. Below, all triangles have area 3. Define a Perfect Pick as a dissection into non-congruent lattice-vertexed triangles of the same area.

Triangles of area 3

What is the largest possible smallest angle within a a Perfect Pick square dissection?