Name of theorem about two quadrilaterals with parallel edges

Question

I'm looking for a name for the following theorem:

If $abAB$ lie on one line and $cdCD$ lie on another line, and furthermore $ac\Vert AC,ad\Vert AD,bc\Vert BC$, then $bd\Vert BD$.

One can use Desargues' theorem to prove this, which is why $O,e,E$ are included in the figure. I know that in Germany, the theorem is sometimes called Scherensatz, which literally translates to scissors theorem. Theorem names often aren't translated literally, though, so I'm unsure. And Wikipedia doesn't seem to know this theorem at all.

Answer 1

In the book The Four Pillars of Geometry it is said:

The second consequence of Desargues theorem is called the "scissors theorem". I do not know how common this name is, but it is used on p. 69 of the book Fundamentals of Mathematics II. Geometry, edited by Behnke, Bachmann, Fladt, and Kunle. In any case, it is an apt name, as you will see from Figure 6.13.

So, as you correctly stated, this theorem is rightfully called scherensatz. I don't know who first proved the theorem, or who was the first to coin this term, but for certain this result appears in Foundations of Geometry by Hilbert, without a name, and as scherensatz in the book by Behnke et. al.

Answer 2

Figures are similar and similarly placed in Geometric similitude, Scaling of figures around a common pole O to Zoom or Reduce.