You could use the "proportion theorem" in Euclidean Geometry: A line drawn parallel to one side of a triangle divides the other sides proportionally.
Folding the paper in quarters, draws three parallel lines, equal distances apart. Folding the cross section from a corner to the third fold, creates two right angle triangles, with two lines parallel to the third side - the side we want to divide in three.
Consider the triangle whose sides are both on the edge of the paper: the parallel lines are still the same distance apart which means the one right angle side is divided into three equal parts (thirds). Using the proportion theorem, we can conclude that the hypotenuses is now also being divided into thirds by the parallel "lines".
Finally we fold perpendicular lines towards the side we want to divide through these intersection points. These lines are again parallel to a side in this triangle therefor dividing the the other side proportionally, which is again in thirds.
By extension you can then fold a paper into n fold by folding it in half m times so that $2^m>n$ then folding the cross section to the n-th parallel lines and then folding the n perpendiculars that will divide the side as desired.