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See Wikipedia 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, one can also consider partition of a quadrilateral into triangles of equal perimeter and equal diameter.

YCor
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Nandakumar R
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I showed in my (still unanswered) question 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.

Johan Wästlund
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  • Thank you very much for those definitive answers to the main area question. That leaves the equal perimeter and diameter dissection into triangles; the equal perimeter case might be complicated - indeed, as claimed in a comment on https://math.stackexchange.com/questions/2822589/dissect-square-into-triangles-of-same-perimeter, "A dissection of a rectangle into seven triangles of equal perimeter has been demonstrated". – Nandakumar R Jun 07 '21 at 14:31