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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