Triangle Parenting

Geometry Level 5

Let $$\bigtriangleup_1, \bigtriangleup_2, \ldots, \bigtriangleup_n$$ be similar triangles (either directly or not) on a plane. The triangle $$\bigtriangleup_i$$ has vertices named $$A_i$$, $$B_i$$, $$C_i$$ at random. We call these parent triangles.

Every pair of parent triangles $$\left(\bigtriangleup_x,\bigtriangleup_y\right)$$ has exactly one child triangle, which has vertices on the middle points of the segments $$A_xA_y$$, $$B_xB_y$$, $$C_xC_y$$ connecting its parents.

No pair of parents has a degenerate child triangle.

Find the least $$n$$ such that, for every choice of the initial triangles, there exist at least one child triangle similar to its parents.

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