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 |

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