This post is based on <https://math.stackexchange.com/questions/2822589/dissect-square-into-triangles-of-same-perimeter>, [On a possible variant of Monsky's theorem](https://mathoverflow.net/questions/394721/on-a-possible-variant-of-monskys-theorem) and [Cutting convex polygons into triangles of same diameter](https://mathoverflow.net/questions/392596/cutting-convex-polygons-into-triangles-of-same-diameter).


**Question 1:** Is this statement true: "For any convex polygon, there is some finite value of a positive integer *n* such that the polygon allows partition into *n* triangles all of which are of same area."?


**Note:** The above claim, if true, will automatically hold for infinitely many values of *n* - one only needs to subdivide the equal area triangular pieces further into equal area triangles. Further, replacing 'area' with 'perimeter' or 'diameter' above generates further questions.


**Question 2:** Is this claim true: "For any given convex polygon, there is at least one finite value of *n* such that the polygon allows partition into *n* triangles all of *equal area and equal perimeter*"?


**Note:** If question 1 has a negative answer, that would invalidate question 2. And there could be convexity-relaxed versions of both above questions.


**An extension added on 15th August, 2022:** If claim 1 is proved invalid (answer below makes progress in that direction), one could ask a further question: If a convex *m*-gon can be cut into some number *n* of triangles, what is the *lower bound* on the number of distinct areas/perimeters/diameters these n triangles can have - as a function of *m* and *n*?



Partitioning convex polygons into triangles of equal area and perimeter
Question 1 indeed has a negative answer, as previous responders have speculated. See e.g. the 2008 paper by Monsky and Jepsen:


[Constructing Equidissections for Certain Classes of Trapezoids"](https://doi.org/10.1016/j.disc.2007.10.031), Discrete Mathematics, 308 (23): 5672–5681, doi:10.1016/j.disc.2007.10.031, Zbl 1156.51304


(I copied this reference from the Wikipedia page on "equidissections".)


I think the simplest examples they prove have no equidissections are trapezoids $(0,0), (1,0), (0,1), (a,1)$ where $a$ is transcendental.
Interestingly there are still (algebraic) values of $a$ for which it is not known whether or not any equidissections exist.


