Monsky's Theorem - variants

An earlier attempt to ask questions beyond this theorem is here .

Monsky's theorem: it is not possible to dissect a square into an odd number of equal area triangles (Note: this implies no rectangle supports such a dissection).

Question: What about cutting a square into triangles of equal perimeter? For what n can a square be cut into n equal perimeter triangles?

Remark: This discussion shows the dissection of a rectangle into 7 triangles of equal perimeter; so there *might* not be a pattern like Monsky theorem for dissection of a square into triangles of equal perimeter.

Observations: It is easy to see that any triangle can be cut into n triangles of equal area for ANY n and as Monsky proved, a square allows such a partition into equal area triangles only for even n. Moreover, by patching together 3 triangles of equal area that share a vertex and with adjacent triangles sharing an edge, one can form a convex pentagon that can be cut into any n number of triangles all of equal area where n is a multiple of 3.

Question: Can one assert that any convex polygon allows partition into *some* finite number n of equal area triangles? Note: If this claim is true, any convex polygon will allow partition into n equal area triangles for infinitely many different values of n.
Are there convex polygons which support more equal area triangulations than the square and less than the triangle? Are there convex polygons which allow partition into say, only odd numbers of equal area triangles?

Further question: What if we need the triangular pieces all to have equal area and equal perimeter? For example, does this claim hold: Any convex polygon will allow partition into some finite number n of triangles all of same area and perimeter (this is a stronger version of a claim given a little above)?