See [Wikipedia](https://en.wikipedia.org/wiki/Monsky%27s_theorem) for Monsky's theorem which states: it is not possible to dissect a square into an odd number of triangles all of equal area.


**Questions:** Are there quadrilaterals that allow partition into any number of equal area triangles? On the other hand, are there quadrilaterals that cannot be cut into any number of equal area triangles? Basically, one asks for those quads which are most (and least) 'susceptible' to partition into equal area triangles.


**Note:** In the spirit of [this MathSE question](https://math.stackexchange.com/questions/2822589/dissect-square-into-triangles-of-same-perimeter), one can also consider partition of a quadrilateral into triangles of equal perimeter and equal diameter.



On a possible variant of Monsky's theorem
I showed in my (still unanswered) question [Is the equidissection spectrum closed under addition?](https://mathoverflow.net/questions/133777/is-the-equidissection-spectrum-closed-under-addition)
that a "kite" with corners in $(-1,0), (0,-1), (0,1), (2,0)$ can be equidissected into any number $n\geq 2$ of triangles. That should answer the first question. Regarding the second question, I think it is known, and mentioned on Wikipedia, that a trapezoid where a certain parameter is transcendental has no equidissection.


