I was reading through Dijkstra's 'A Collection of Beautiful Proofs' and stumbled upon this elegant piece of work:
11. Reconstructing an odd polygon from the midpoints of its sides.
We shall show the construction for poly = 5.
For the pentagon ABCDE , the points marked AB, BC. CD, DE, and EA respectively are the midpoints of its successive sides. Given the positions of those five midpoints, it is requested to reconstruct the original pentagon ABCDE. [...]
Click here for a representation of the example $^1$
So Dijkstra solves this problem by taking an arbitrary point $X_0$ and rotating it around $AB$, $BC$, $CD$, $DE$ and $EA$ successively, as can be seen in the image. The original point $A$ is the midpoint of the line from the resulting point $X_5$ and the intial, arbitrary point $X_0$. The position of $B$, $C$, $D$ and $E$ indeed follow trivially.
When I read this solution, I thought about other (probably less elegant) ways to reconstruct the polygon - and then how it would be possible to extent this to any arbitrary (odd) polygon.
In the case of a triangle it seems pretty simple and has been answered here already: Is it possible to reconstruct a triangle from the midpoints of its sides?
I am not that familiar with Mathematics and Geometry anymore, but what would be another or more general solution for the above described problem?
$^1$ Picture and excerpt taken from: https://www.cs.utexas.edu/users/EWD/transcriptions/EWD05xx/EWD538.html