Is this converse results of Varignon's theorem known?

Question

The Varignon's Theorem on Quadrilateral is very well known results of Plane Geometry and we have find the Converse of this theorem on Quadrilateral and generalise this for 2n-sided convex irregular polygon and Wanted to know about refrence, article , website where these given below results are mentioned so that we could verify whether this is known or not .

Result 01: Converse of Varignon Theorem on Quadrilateral :

Let $A_1A_2A_3A_4$ be parallelogram and Let $B_1$ be arbitrary point and reflect $B_1$ in $A_1$ to get $B_2$ and reflect $B_2$ in $A_2$ to get $B_3$ and Simillary reflect $B_3$ in $A_3$ to get $B_4$ then point { $A_4, B_4, B_1$} become Straight line where $A_4$ is midpoint of $B_1B_4$ as Shown in Figure given below:

Note : Sinc Varignon theorem said that midpoint of Line segment of Quadrilateral makes Parallelogram so will take the midpoint of each Line segment of 2n sided convex polygon as discussed in Result 02 given below.

Result 02 (Generalisation of Converse of Varignon theorem) :

Let $A_1A_2A_3.......A_2n$ be $2n$ sided convex irregular polygon and Let {$B_1, B_2,B_3,......, B_2n$} be m($A_1, A_2$) ; m($A_2, A_3$) ;m($A_3, A_4$) ;m($A_4, A_5$) ;.......... ;m($A_2n, A_1$).

Let $C_1$ be arbitrary Point anywhere in Geometry plane and

reflect $C_1$ in $B_1$ to get $C_2$;

reflect $C_2$ in $B_2$ to get $C_3$;

reflect $C_3$ in $B_3$ to get $C_4$;

Reflect $C_4$ in $B_4$ to get $C_5$ and Simillary defined {$C_6,C_7, C_8,C_9,...., C_2n$ } cyclically then point {$C_2n, B_2n, C_1$} will become Straight line and $B_2n$ become m($C_2n, C_1$).

*See figure for $n=4$:

Note:

(1) Here symbol m(A, B) denotes midpoint of Line Segment AB.

(2)point $A_1, A_2,........, A_2n$ are arbitrary Point.

Answer 1

In the general case, segments $\,A_1A_2\,$ and $\,C_1C_2\,$ have the same midpoint $\,B_1\,$, so $\,A_1C_2A_2C_1\,$ is a parallelogram and therefore $\,\overrightarrow{A_1C_1}=-\overrightarrow{A_2C_2}\,$. Repeating the argument for the next pairs:

$$ \overrightarrow{A_1C_1}=-\overrightarrow{A_2C_2}=\overrightarrow{A_3C_3}=-\overrightarrow{A_4C_4}=\dots=-\overrightarrow{A_{2n}C_{2n}} $$

$\overrightarrow{A_1C_1}=-\overrightarrow{A_{2n}C_{2n}}$ means that $A_{2n}C_{2n}A_1C_1$ is a parallelogram, so the diagonals $\,A_{2n}A_1\,$ and $\,C_{2n}C_1\,$ have the same midpoint, in other words $\,C_{2n}\,$ and $\,C_1\,$ are symmetric about $\,B_{2n}\,$.

The property holds true for any $\,2n\,$ points $\,A_1, A_2, \dots, A_{2n}\,$, not necessarily forming a convex polygon, and not necessarily even in the same plane.

Don't know that this property has a name (and it's not a direct converse of Varignon's theorem either), but related questions have been asked before, for example Midpoint polygons?.

Answer 2

This is known that the midpoint of segment of polygon is varignon polygon as shown in this reference but the converse result could be new as I don't able to find it anywhere.

Also see the Extension of Varignon theorem here and Also see it's other generalisation on 2n sided polygon here

Note: for more extension , you can see the reference mention in these two above link.