Nested triangles

We'll name "nested triangles" a sequence of triangles, the vertices of which lie on the sides of the next one.
We note PQR ∈ ABC to say that P lies on AB, Q on BC and R on CA.
Two triangles are said "parallel" if their corresponding sides are parallel.
Nested triangles build a "parallel sequence" if all triangles with same parity are parallel.

This being said, prove that, in any parallel sequence of nested triangles, the areas build a geometric sequence.
Specifically with three nested triangles PQR ∈ ABC ∈ VTU and PQR // TUV, area of ABC is geometric mean between areas of PQR and TUV

 Area(PQR)Area(TUV) = Area(ABC)² 
Hint

Details

With aknowledgement to Rainer Rosenthal, who found this topic as "Babuschka Dreiecke" during his search of a AHA solution to minimum inscribed triangle.

 

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