The quatrilateral connecting the middles of the edges of a quatrilatera

A 3d quatrilateral is defined by 4 points:

#declare P  = < 1.5, 0.0,-4.0>;
#declare Q  = < 4.0, 1.0, 3.5>;
#declare R  = < 0.0, 1.5, 3.5>;
#declare S  = <-3.0, 4.0,-2.5>;

A 3d quatrilateral need not to be flat! In contrast to a 2d quatrilateral the area of a 3d quatrilateral cannot be defined unambiguous by 4 points!

A 3d quatrilateral need not to be flat!

The middle of the edges M_PQ, M_QR, M_RS und M_SP of the quatrilateral are to calculate as follows:

#declare M_PQ = (P+Q)/2;
#declare M_QR = (Q+R)/2;
#declare M_RS = (R+S)/2;
#declare M_SP = (S+P)/2;

If we connect the middles of the edges, we get a parallelogram.
Also if we interchange i.e. the points S and R:

#declare S  = < 0.0, 1.5, 3.5>;
#declare R  = <-3.0, 4.0,-2.5>;

again - we get a parallelogram.


The quatrilateral connecting the middles of the edges of a quatrilatera is always a parallelogram!

Proof :
For the vector of the edge M_PQM_QR we have:
M_PQM_QR = (Q+R)/2 - (P+Q)/2
               = Q/2 + R/2 - P/2 - Q/2
               = R/2 - P/2. (I)

For the vector
of the edge opposite M_PSM_RS we have:
M_PQM_QR = (R+S)/2 - (S+P)/2
               = R/2 + S/2 - S2 - P/2
               = R/2 - P/2. (II)

Comparing (I) with (II) we see, that both edges are represented by the same vector, that means they are parallel and they have the same length.
A quatrilateral with two opposite parallel edges of the same length must be a parallelogram.

The quatrilateral connecting the middles of the edges of a quatrilatera is always a parallelogram.
This scene for POV-Ray: ".txt" file or ".pov" file

The quatrilateral connecting the middles of the edges of a quatrilatera with interchanged Points S and R.
This scene for POV-Ray: ".txt" file or ".pov" file