For an integer $n \ge 3$ and positive integer $m$, is it possible to divide a triangle into $m$ $n$-gons of equal area?
For the $3$-gon, or the triangle, you can divide an edge into $m$ equal segments, then draw lines from all dividers of those segments to the opposite vertex. (I'm not sure how to draw diagrams.)
I have no idea how to even begin with other polygons. Can someone give a hint? Thanks in advance.
Let us first notice that each triangle may be divided into two quadrangles of equal area. We choose the longest edge as a base to a trapezoid and choose the second base of the trapezoid such that its area is equal to half the area of the triangle.
Moreover, we can also divide a triangle into three quadrangles of equal areas - we choose a point inside a triangle and then three points on the edges - we may freely move the points such that the areas of resulting quadrangles will be pairwise equal.
We can now proceed to a draft of proof for $m\geq 2$ and $n=4$, as I believe it provides a base toward generalization. Let us first assume that $m$ is even - we could then divide the big triangle into $\frac{m}{2}$ triangles of equal areas and then each small triangle into two quadrangles of equal areas - e.g. in a way presented above. The problem arises when $m$ is odd. However, let us notice that every natural number $m > 1$ may be represented as $$m = 3x + 2y$$ for some natural numbers $x$ and $y$. Let us divide an edge of the triangle into $x+y$ segments of lengths $k$ and $l$, such that $k=\frac{3}{2}l$. If we drew lines from the endpoints of the segments to the opposite vertex of the triangle, we would achieve $x$ "bigger" triangles and $y$ "smaller" triangles, such that the area of a bigger triangle is equal to $\frac{3}{2}$ of the area of the smaller triangle. We now proceed to divide the $x$ bigger triangles each into three quadrangles of equal areas and divide each of $y$ smaller triangles into two quadrangles of equal areas. We have therefore achieved $3x+2y=m$ quadrangles having pairwise equal areas.
I believe the proof above may be generalized for $n>4$ if we started with $n=4$ and then put additional vertices on some edges between quadrangles. We then achieve a degenerated dissection into pentagons and we should be able to move the vertices "slightly" only to lose the degeneracity.