From the Iranian Geometry Olympiad, 2017:
Each side of square $ABCD$ with side length of $4$ is divided into equal parts by three points. Choose one of the three points from each side, and connect the points consecutively to obtain a quadrilateral. Which numbers can be the area of this quadrilateral? Here's a construction.
I am interested in finding out:
- How many quadrilaterals of distinct dimensions can you draw?
- How many quadrilaterals of distinct areas can you draw?
A naïve approach is to multiply the number of possible vertices on each side of the square ($3 \cdot 3 \cdot 3 \cdot 3$). But that would overcount the number of distinct quadrilaterals (since you could obtain one by rotating or reflecting the other).
Of course, in the original problem, a brute-force approach would have sufficed. I would like to find the solution for a general case: given a square of $n + 1$ side length, how many distinct quadrilaterals are possible?