Reverse triangulation

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I have a polygon (rhombus in this example)

diamond

Let's assume we have some arbitrary triangulation for that with N triangles (4 of them here).

The question is the following: having 4 triangles in the first place (see picture below)

initial triangles

can I build a desired shape using them? Move and rotation is allowed (but not scaling). Is there an algorithm or any approach that could help validate this statement?

The only thing I have in mind is to compare areas (sum of triangles and the polygon) but that doesn't give much. Thanks for any suggestion.