Here is a geogebra file that I did: https://www.geogebra.org/calculator/vkteyrf2
- $ABC$ is a triangle
- $A_1 \in BC$
- $B_1 \in AC$
- $C_1 \in AB$
- $M \in BC$ is a moving point from which I drew paralells to each side of the triangle. The parallel that I drew from $M$ intersects the $AB$ side in point $M_1$ after that from this new $M_1$ point I drew a parallel with $C_1B_1$ and I got a new intersection point let's say $M2$ I repeated this for 5 times and the construction closed.
From what I can tell this construction will become closed if $A_1,B_1,C_1$ are midpoints, how can I prove this and does this construction work for any polygons?
I appreciate any tips or help regarding this problem!